Robust Aspects of Hedging and Valuation in

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Robust Aspects of Hedging and Valuation in Incomplete Markets and related Backward SDE Theory

DISSERTATION zur Erlangung des akademischen Grades Doctor rerum naturalium (Dr. rer. nat.) im Fach: Mathematik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakult¨at der Humboldt-Universit¨at zu Berlin von M.Sc. Kl´ ebert, Kentia Tonleu Pr¨asident der Humboldt-Universit¨at zu Berlin Prof. Dr. Jan-Hendrik Olbertz Dekan der Mathematisch-Naturwissenschaftlichen Fakult¨at Prof. Dr. Elmar Kulke

Gutachter:

1. Prof. Dr. Dirk Becherer 2. Prof. Dr. Romuald Elie 3. Prof. Dr. Peter Imkeller

Tag der Einreichung: Tag der m¨undlichen Pr¨ufung:

31.07.2015 04.12.2015

Abstract This thesis studies backward stochastic differential equations (BSDEs), and robust notions of dynamic good-deal valuation and hedging in incomplete financial markets. We start by a mathematical theory, concerning the analysis of BSDEs with jumps driven by random measures that can be of infinite activity with time-inhomogeneous compensators. Under concrete conditions that are easy to verify in practical applications, we provide existence, uniqueness and comparison results for bounded solutions for a class of generator functions that are not required to be globally Lipschitz in the jump integrand. To illustrate the ease of applicability of our results, we solve the exponential and power utility maximization problems with additive and multiplicative liability respectively. The rest of the thesis deals with the more application-oriented problem of robust valuation and hedging in incomplete markets. We are concerned with the no-good-deal approach, which computes good-deal valuation bounds by using as pricing measures only a subset of the risk-neutral measures satisfying a constraint on the Girsanov kernels described by correspondences with economic meaning. Examples of such constraints are given by bounds on instantaneous Sharpe ratios, optimal growth rates, or expected utilities. Throughout we study a notion of good-deal hedging that corresponds to good-deal valuation, and for which hedging strategies arise as minimizers of some dynamic coherent risk measures allowing for optimal risk sharing with the market. Hedging is shown to be at least mean-self-financing in the sense that tracking (hedging) errors satisfy a supermartingale property under suitable a-priori valuation measures. The latter is then interpreted as robustness of good-deal hedging, with respect to the family of valuation measures as generalized scenarios. We derive constructive results on good-deal valuation and hedging using BSDEs. The results are obtained in a jump framework with unpredictable event-risk, as well as in a Brownian setting with model uncertainty. In the jump framework we use the theory on BSDEs with jumps, and provide examples in (semi-)Markovian models, which are particularly relevant for actuarial applications. In the Brownian setting, we provide new examples for concrete no-good-deal constraints, with closed-form expressions for valuations and hedges given via classical option pricing formulas (Black-Scholes, Margrabe or Heston). Moreover, under Knightian uncertainty (ambiguity) about the objective real-world probability measure which is not taken to be precisely known, we study robustness of hedging taking into account the investor’s aversion towards ambiguity. Assuming multiple reference priors as candidates for the (uncertain) real-world measure, a worst-case approach leads to good-deal hedging that is robust with respect to uncertainty in the sense that it is at least mean-self-financing uniformly over all priors. Results are presented for drift uncertainty and volatility uncertainty separately, using classical BSDEs for the former and second-order BSDEs for the latter. Under drift uncertainty, we also show existence of a worst-case prior with respect to which dynamic valuations and hedges can be computed like in the absence of uncertainty. Here the robust approach yields that good-deal hedging is equivalent to risk-minimization with respect to a suitable measure if drift uncertainty is sufficiently large. In the case of volatility uncertainty, we provide an example for put options in an uncertain volatility model of Black-Scholes’ type, where explicit solutions for (robust) good-deal valuations and hedges are computable under a worst-case prior.

Zusammenfassung Diese Arbeit untersucht stochastische R¨ uckw¨artsdifferentialgleichungen (BSDEs) und robuste Konzepte von dynamischer Good-Deal-Bewertung und -Hedging in unvollst¨andigen Finanzm¨arkten. Wir beginnen mit einer mathematischen Theorie zur Analyse von BSDEs mit Spr¨ ungen, getragen von zuf¨alligen Maßen, die von unendlicher Aktivit¨at mit zeitlich inhomogenem Kompensator sein k¨onnen. Unter konkreten Bedingungen, die in praktischen Anwendungen leicht zu verifizieren sind, liefern wir Existenz-, Eindeutigkeits- und Vergleichsergebnisse beschr¨ankter L¨osungen f¨ ur eine Klasse von Generatorfunktionen, welche nicht-notwendigerweise global Lipschitz-stetig im Sprungintegranden sein m¨ ussen. Wir l¨osen das Maximierungsproblem f¨ ur exponentiellen Nutzen bei additiver Verbindlichkeit und f¨ ur Power-Nutzen bei multiplikativer Verbindlichkeit, um die Anwendbarkeit unserer Resultate zu veranschaulichen. Der u ¨brige Teil der Arbeit besch¨aftigt sich mit dem eher anwendungsorientierten Problem der robusten Bewertung und des Hedgings in unvollst¨andigen M¨arkten. Wir befassen uns mit dem No-Good-Deal-Ansatz, welcher Good-Deal-Grenzen liefert, indem als Bewertungsmaße lediglich eine Teilmenge der risikoneutralen Maße betrachtet werden, die eine Bedingung an den GirsanovKern – beschrieben durch Korrespondenzen mit ¨okonomischer Bedeutung – erf¨ ullen. Beispiele solcher Bedingungen sind Grenzen f¨ ur instantanen Sharpe-Ratio, optimale Wachstumsrate oder erwarteten Nutzen. Durchweg untersuchen wir ein Konzept des Good-Deal-Hedgings, das GoodDeal-Bewertung entspricht und f¨ ur welches Hedgingstrategien als Minimierer geeigneter dynamischer koh¨arenter Risikomaße auftreten, was optimale Risikoteilung mit der Markt erlaubt. Wir zeigen, dass Hedging mindestens im-Mittel-selbstfinanzierend ist. Das heißt, dass Hedgefehler unter geeigneten A-priori-Bewertungsmaßen eine Supermartingaleigenschaft haben. Dies wird als Robustheit des Good-Deal-Hedgings bez¨ uglich der Familie von Bewertungsmaßen, gesehen als verallgemeinerte Szenarien, interpretiert. Wir leiten konstruktive Ergebnisse zu Good-Deal-Bewertung und -Hedging mittels BSDEs her. Die Ergebnisse werden sowohl im Rahmen von Prozessen mit Spr¨ ungen mit unvorhersehbarem Ereignisriungen nutzen siko, als auch im Brown’schen Rahmen mit Modellunsicherheit erzielt. Im Falle von Spr¨ wir die Theorie zu BSDEs mit Spr¨ ungen und liefern Beispiele in (Semi-)Markov-Modellen, die insbesondere f¨ ur versicherungsmathematische Anwendungen von Bedeutung sind. Im Brown’schen Fall liefern wir neue Beispiele f¨ ur konkrete No-Good-Deal-Bedingungen mit expliziten Formeln f¨ ur Bewertung und Hedging, aufbauend auf klassischen Optionsbewertungsformeln (Black-Scholes, Margrabe oder Heston). Unter Knight’scher Unsicherheit bez¨ uglich des nicht genau bekannten objektiven realen Maßes untersuchen wir hier Robustheit des Hedgings unter Ber¨ ucksichtigung der Abneigung des Investors gegen Ungewissheiten. Bei Annahme mehrerer Referenzmaße als Kandidaten f¨ ur das (unsichere) reale Maß f¨ uhrt ein Worst-Case-Ansatz zu Good-Deal-Hedging, welches robust bez¨ uglich Unsicherheit, im Sinne von gleichm¨aßig u¨ber alle Referenzmaße mindestens im-Mittel-selbstfinanzierend, ist. Die Ergebnisse zu Drift- und Volatilit¨atsunsicherheiten werden separat pr¨asentiert, wobei f¨ ur erstere klassische BSDEs und f¨ ur letztere BSDEs zweiter Ordnung zur Anwendung kommen. Bei Driftunsicherheit zeigen wir außerdem Existenz eines Worst-Case-Maßes unter dem sich Bewertungen und Hedging wie bei Abwesenheit der Unsicherheit berechnen lassen. Hier liefert der Robustheitsansatz, dass bei hinreichend großer Driftunsicherheit Good-Deal-Hedging ur ¨aquivalent ist zur Risikominimierung. Im Falle von Volatilit¨atsunsicherheit legen wir ein Beispiel f¨ Put-Optionen in einem Black-Scholes-artigen Modell mit unsicherer Volatilit¨at vor, in dem explizite L¨osungen zur (robusten) Good-Deal-Bewertung und Hedging unter einem Worst-Case-A-priori-Maß berechnet werden k¨ onnen.

Acknowledgements First, I am extremely grateful to my advisor Dirk Becherer for introducing me to the fascinating areas of stochastic analysis and mathematical finance, and for patiently guiding me throughout the completion of this work. His support, ideas, discerning comments and good sense of mathematical writing were indispensable to the development and writing of this work, and greatly contributed to my scientific training as a whole. In particular I thank him for being such a kind person, for always being willing to share his knowledge with me and for providing advices for my diverse initiatives as well academically as on a personal level. I would like to thank Romuald Elie and Peter Imkeller for immediately agreeing to be co-examiners of this thesis. I also thank my colleagues and friends in Berlin, for the fruitful discussions and numerous extracurricular activities that made the process of completing this thesis quite an agreeable experience. In particular I think about Achref Bachouch, Julio Backhoff, Todor Bilarev, Martin B¨ uttner, Peter Frentrup, Guanxing Fu, Paulwin Graewe, Elena Ivanova, Martin Karliczek, Victor Nzengang, Ludovic Tangpi, and many more that I have forgotten to mention and to whom I am greatly grateful. Special thanks are addressed to Axel Mosch and Guillaume Richard for help with numerical simulations in the examples in Chapter 3. Support from the Humboldt-Universit¨at zu Berlin, and from the German Science Foundation DFG via the Berlin Mathematical School and the Research Training Group 1845 Sto-A is gratefully acknowledged. They generously provided financial support and a propitious research environment inherent to the completion of this work. Most special thanks go to my parents and siblings for their long-term regular support and for bearing with me leaving home to continue my studies abroad. Finally, I dedicate this thesis to Darl`ene and Keynane for their love and moral assistance. They have been more than a source of motivation and perseverance, especially during the last months preceding the submission of the thesis.

iv

` mes deux amours, Darl`ene et Keynane. A

Contents Introduction

1

1 Concrete criteria for wellposedness and comparison of BSDEs with jumps of infinite activity 19 1.1

Mathematical framework and preliminaries . . . . . . . . . . . . . . . . . . . . 19

1.2

Comparison theorems and a-priori-estimates . . . . . . . . . . . . . . . . . . . 25

1.3

Existence and Uniqueness of bounded solutions . . . . . . . . . . . . . . . . . 32

1.4

1.3.1

The case of finite activity . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.3.2

The case of infinite activity . . . . . . . . . . . . . . . . . . . . . . . . 34

Applications to optimal control problems in finance . . . . . . . . . . . . . . . 43 1.4.1

Exponential utility maximization . . . . . . . . . . . . . . . . . . . . . 45

1.4.2

Power utility maximization . . . . . . . . . . . . . . . . . . . . . . . . 52

2 Hedging under generalized good-deal bounds in jump models with random measures 55 2.1

Mathematical framework and preliminaries . . . . . . . . . . . . . . . . . . . . 55

2.2

Good-deal valuation and hedging . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.3

Case of uniformly bounded correspondences . . . . . . . . . . . . . . . . . . . 65

2.4

2.5

2.3.1

Results for constraint on instantaneous Sharpe ratios (bounded case) . 69

2.3.2

Results for ellipsoidal constraint and uncertainty about jump intensities

Case of non-uniformly bounded correspondences . . . . . . . . . . . . . . . . . 78 2.4.1

Results for constraint on instantaneous Sharpe ratios (unbounded case)

2.4.2

Results for constraint on optimal expected growth rates . . . . . . . . . 81

81

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3 Hedging under generalized good-deal bounds and drift uncertainty 3.1

76

99

Mathematical framework and preliminaries . . . . . . . . . . . . . . . . . . . . 99 viii

3.2

3.3

3.4

3.1.1

Parametrizations in an Itˆo process model . . . . . . . . . . . . . . . . . 102

3.1.2

Good-deal valuation with uniformly bounded correspondences . . . . . 104

3.1.3

Good-deal valuation with non-uniformly bounded correspondences . . . 105

Dynamic good-deal hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.2.1

Results for ellipsoidal no-good-deal constraints . . . . . . . . . . . . . . 110

3.2.2

Examples for good-deal valuation and hedging with closed-form solutions113

Good-deal valuation and hedging under model uncertainty . . . . . . . . . . . 120 3.3.1

Model uncertainty framework . . . . . . . . . . . . . . . . . . . . . . . 121

3.3.2

No-good-deal constraint and good-deal bounds under uncertainty . . . 122

3.3.3

Robust approach to good-deal hedging under model uncertainty . . . . 125

3.3.4

Hedging under model uncertainty for ellipsoidal good-deal constraints . 125

3.3.5

The impact of model uncertainty on robust good-deal hedging . . . . . 131

3.3.6

Example with closed-form solutions under model uncertainty . . . . . . 134

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

4 Hedging under good-deal bounds and volatility uncertainty: a 2BSDE approach144 4.1

4.2

4.3

Mathematical framework and preliminaries . . . . . . . . . . . . . . . . . . . . 144 4.1.1

The local martingale measures . . . . . . . . . . . . . . . . . . . . . . 144

4.1.2

Spaces and norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.1.3

Second order backward stochastic differential equations . . . . . . . . . 147

Market model and good-deal constraint under volatility uncertainty . . . . . . . 155 4.2.1

Financial market with volatility uncertainty . . . . . . . . . . . . . . . 156

4.2.2

No-good-deal constraint . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Good-deal bounds and hedging under volatility uncertainty . . . . . . . . . . . 159 4.3.1

Good-deal bounds under volatility uncertainty . . . . . . . . . . . . . . 160

4.3.2

Robust good-deal hedging under volatility uncertainty . . . . . . . . . . 163

4.3.3

Example for options on non-traded assets . . . . . . . . . . . . . . . . 169

ix

References

176

List of Figures

190

List of Tables

191

x

Introduction This thesis is concerned with backward stochastic differential equations (BSDEs) and with hedging and valuation of contingent claims in incomplete financial markets. BSDEs have by now found numerous applications in mathematical finance, where they have proved to be suitable tools in describing solutions to many stochastic optimization problems of practical relevance. BSDEs form a common theme for all chapters of the thesis, and will be used throughout in different forms depending on the problem at hand. In particular, Chapter 1 of the thesis is concerned with theoretical foundations of BSDEs with jumps (in short JBSDEs), which are BSDEs driven jointly by a Brownian motion and a random measure. We study wellposedness (existence and uniqueness) and comparison for bounded solutions to this class of BSDEs, for jumps that may have infinite activity with compensators being possibly time-inhomogeneous. Moreover in this chapter, applications of the JBSDE theory will be presented dealing with the utility maximization problem in finance. The remaining chapters of the thesis (Chapters 2 to 4) deal overall with the problem of valuation and hedging of contingent claims in incomplete financial markets. Valuation and hedging are classical topics in mathematical finance for which many approaches have been studied in the literature, especially in the context of incomplete markets where some risks may not be perfectly hedgeable, and valuation and (partial) hedging may involve solving stochastic optimization problems. As far as this thesis is concerned, we will focus on the no-good-deal approach, which does not only prevent arbitrage opportunities from the market, but also excludes an economically meaningful notion of deals that are “too good”. This leads to so-called good-deal valuation bounds, to which a corresponding concept of hedging will be associated. In a general framework with no-good-deal constraints defined in terms of abstract correspondences (set-valued mappings) for the pricing measures, we will obtain results on good-deal hedging and valuation in terms of solutions to BSDEs. We will provide examples with explicit formulas that facilitate computations, for specific correspondences associated to more concrete no-good-deal constraints. In particular in Chapter 2 we will apply our theoretical results on JBSDEs from Chapter 1 to good-deal valuation and hedging in a setup allowing for unpredictable event-risk, which is modelled by a discontinuous filtration supporting a random measure and a Brownian motion simultaneously. Another topic of central interest in this thesis is robustness. In general, a robust concept will refer to one which remains effective under different admissible market scenarios/variables. In the presence of model uncertainty (ambiguity), the scenarios will correspond to the uncertain priors (models) and we will analyze in Chapters 3 and 4 robust concepts of good-deal valuation and hedging with respect to model uncertainty. We will focus on continuous filtrations, which will allow us to use the classical theory of BSDEs driven solely by a Brownian motion for the case of uncertainty about the excess return of traded assets (cf. Chapter 3), and the theory of second-order BSDEs (shortly 2BSDEs) for the case of uncertainty about the volatility (cf. Chapter 4). Before giving a more

1

Page 2 detailed account of the contributions of the thesis, we next explain the necessary background on BSDEs, valuation and hedging in incomplete markets, and model uncertainty. These three general themes are central to the thesis and their connections with different chapters will also be made more precise.

An overview of the theory of backward SDEs BSDEs are studied and used intensively in this thesis. To relate the contributions of the thesis to the historical developments of BSDEs, we present a short overview of advances in the theory that culminated in a wide range of applications to optimal control problems in mathematical finance. Classical BSDEs form a class of stochastic differential equations (SDEs) of the type Z

Yt = ξ + t

T

fs (Ys , Zs ) ds −

Z

T

Zs dWs ,

t ∈ [0, T ],

t

They are described by a semimartingale dynamics for which a terminal condition ξ is given (instead of an initial one as for forward SDEs), and the generator function f (drift of the dynamics) varies with the value process Y of the equation and the control process Z integrated by the driving Brownian motion W under its natural filtration (Ft )t≤T . The solution of a BSDE consists of the couple (Y, Z). Originally BSDEs appeared in [Bis73] with linear generators functions. [PP90] were the first to study existence and uniqueness of square integrable solutions in the classical setting for BSDEs under global Lipschitz assumptions on the generator. Such BSDEs will appear in Chapter 3, under a uniformly boundedness assumptions on the no-good-deal constraint correspondence for the Girsanov kernels of pricing measures. For a detailed exposition on applications of classical BSDEs in mathematical finance and additional results including a comparison principle, we refer to [EPQ97]. Beyond the Lipschitz setting, notable extensions include the case of generators with quadratic growth in the Brownian integrand Z for which [Kob00] has studied bounded solutions (see also [Tev08, BE13]). This has found crucial applications in utility maximization in incomplete markets initiated by [RE00] and [HIM05]. It has been shown in [DHB11] that BSDEs with generators that are of super-quadratic growth are typically illposed. Beyond quadratic growth and with generators that are only convex, [DHK13, DHK15] proved existence and uniqueness of minimal supersolutions relying on compactness rather than fixed-point arguments. This solution concept will be used in Chapter 3, where we will consider no-good-deal constraint correspondences that are not necessarily uniform bounded. Let us mention that by now there exists a plurality of numerical methods for simulation of BSDEs, including Monte-Carlo methods which are particularly relevant for higher dimensional problems. For advances in this direction, we refer to [BT04, GLW05, BD07, GT15, BT14]. BSDEs that are driven not only by a Brownian motion W but additionally by a random measure

Page 3 are shortly referred to as JBSDEs (i.e. BSDEs with jumps) and involve a second stochastic integral with respect to the compensated random measure. Their dynamics are of the form Z

Yt = ξ + t

T

fs (Ys− , Zs , Us ) ds −

Z t

T

Zs dWs −

Z TZ

Us (e) µ ˜(ds, de), t

t ∈ [0, T ],

E

with µ ˜ = µ − ν P denoting the compensated random measure of some integer-valued random measure µ on a space E for a stochastic basis (Ω, FT , (Ft )t≤T , P ). The solution of a JBSDE is now a triple (Y, Z, U ), where the jump integrand U lives in a possibly infinite-dimensional function space and also appears in the generator of the BSDE. For such JBSDEs, [TL94, BBP97] studied square integrable solutions under global Lipschitz conditions in a time-homogeneous setting for Poisson random measures. Bounded solutions to JBSDEs have been studied in [Bec06] for a random measure that is possibly inhomogeneous in time but of finite jump activity, covering a family of generators that satisfy a certain monotonicity property but need not be (globally) Lipschitz in the jump integrand, see also [Par97, Roy06]. A similar study will be considered in Chapter 1 but for possibly infinite activity of jumps, and increased degree of complexity of the generators also allowing for a comparison principle for such JBSDEs. Indeed it appears here (see also [BBP97, Roy06, CE10]) that comparison principles for JBSDEs require more delicate technical conditions than in the Brownian case. These comparison principles will be applied in Chapter 2 to derive JBSDEs for good-deal valuation bounds and associated hedging strategies. We will consider in Chapter 1 applications of our JBSDE theory to the utility maximization problem in finance, for jumps of infinite activity. Note that JBSDEs with generator of quadratic growth in the Brownian integrand have been studied for a particular generator and infinite activity of jumps in [Mor09, Mor10], in [KTPZ15a] also under time-inhomogeneity, and in [EMN14] in general under finite activity assumptions. For numerical analysis of JBSDEs, see e.g. [BE08]. There is a strong connection between BSDEs and the theory of partial differential equations (PDEs). In fact (first-order) Markovian BSDEs for which the generator additionally depends on the solution of a forward SDE, hence referred to as forward-backward SDEs (alternatively FBSDEs), are probabilistic representations `a la Feymann-Kac for second-order quasi-linear PDEs (i.e. PDEs involving only a linear dependency in the Hessian of the solution). Indeed, the PDE terms depending on the second-order derivative of the solution can only arise from the quadratic variation of the forward process via Itˆo’s formula. Probabilistic representations of PDEs pave the way to numerical Monte-Carlo schemes for simulation of their solutions, which again are more relevant for PDEs with high dimensional state-space. Note that in the case of Markovian JBSDEs, an additional integral appears in the formulation of the PDE, hence yielding a partial-integro differential equations (PIDEs). Due to their importance in practice, one would also like as for quasi-linear PDEs to have a probabilistic representation for fully nonlinear PDEs (i.e. PDEs involving a nonlinear dependency in the Hessian of the solution), which are an important class containing e.g. Hamilton-Jacobi-Bellman (HJB) equations. It is

Page 4 exactly this fact that motivated [CSTV07] to formally introduce the notion of 2BSDE originally in connexion to the solution to the second order stochastic target problem first introduced by [ST09]. To ease exposition, a 2BSDE in its simplified form is an equation of the type Z

Yt = ξ− t

T

1 bs −Hs (Ys , Zs , Γs ) ds− Γs a 2

Z t

T

Zs dBs +KT −Kt ,

P -a.s., t ∈ [0, T ], ∀ P ∈ PH

b is the (ω-wise) density of the quadratic variation of the coordinate process B on the where a canonical Wiener space of continuous paths, PH is a subset of (typically mutually singular) local martingale measures for B, and K is a non-decreasing process with K0 = 0. Note that contrary to classical BSDEs, the dynamics of 2BSDEs is required to hold almost-surely under P , for all P in a family PH of reference probability measure, that is to say quasi-surely with respect to PH . In this form the solution to the 2BSDE is the triple (Y, Z, Γ), and the generator Fb is the convex conjugate of a nonlinear function H in its third argument Γ, bs − Hs (Ys , Zs , Γs ). For classical BSDEs, the solution components satisfying Fbt (Yt , Zt ) := 12 Γs a Y and Z correspond to the PDE solution and its gradient (first-order derivative) respectively. For 2BSDEs in a Markovian setting with a canonical forward process, one has an additional unknown variable Γ in the dynamics of the BSDE which essentially corresponds to the Hessian (second-order derivative) of the solution to a fully nonlinear PDE (justifying the appellation “second-order” BSDE). For first applications of 2BSDEs in mathematical finance, let us mention among many others [C ¸ ST07, ST09] for the super-replication problem under Gamma constraint and [MPZ15] for the robust utility maximization under volatility uncertainty [ALP95, Lyo95]. In Chapter 4, we extend the list of applications in finance by using 2BSDEs to describe the solutions to good-deal valuation and hedging problem that are robust (in some sense to be made precise later) with respect to volatility uncertainty. The original formulation of 2BSDEs in [CSTV07] was in a Markovian setting and is somewhat different to the one presented above. The above is a particular case of [STZ12] who used the quasi-sure analysis of [DM06] to obtain a general formulation for possibly non-Markovian 2BSDEs and obtained a wellposedness theory for 2BSDEs with Lipschitz generators. Note that in the language of G-stochastic calculus of [Pen10], wellposedness of 2BSDEs with zero generators can be viewed as a martingale representation theorem for G-martingales (and G-expectations in particular). The wellposedness theory was later extended by [PZ13] to 2BSDEs with quadratic generators. Subsequently, [MPZ13] and [KTPZ15c, KTPZ15b] studied 2BSDEs reflected on an obstacle and 2BSDEs with jumps respectively. Some numerical schemes for 2BSDEs based on Monte-Carlo or/and finite difference methods have been suggested in the literature, e.g. in [CSTV07, FTW08, GZZ15, PT14]; see also [BET09] for a survey on the probabilistic numerical methods for nonlinear PDEs in general.

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Hedging and pricing approaches in mathematical finance Investing in financial markets involves facing some risk that can be synthesized either perfectly (one speaks of replication) or only partially by dynamic trading in liquid assets. The seller of a financial contract (contingent claim) is usually confronted with the following problem: what valuation would she like to sell the claim to the buyer for, to enable a certain form of hedging against the risk of loss at delivery of the claim? A financial market where all contingent claims can be replicated is referred to as complete. The significance of such markets lies in that they allow for pricing by replication so that under the viability of the market the price of a contingent claim is the cost of the replicating portfolio. This was the insight behind [BS73, Mer73] where assuming that asset prices follow a geometric Brownian motion, the authors obtained the price of vanilla call/put options by replicating with the delta-hedging strategy and deriving the celebrated Black-Scholes formula for option pricing. The Black-Scholes formula has been extended for pricing other types of options; for instance, the Margrabe formula [Mar78] is used to price European exchange options, i.e. options to exchange one risky asset for another at a pre-specified maturity time. The Black-Scholes and Margrabe formulas will play a role in the examples of Chapter 3 (see also example at the end of Chapter 4), where we will derive good-deal valuations and hedges via these formulas for incomplete models with traded and non-traded assets. The prices resulting from replication are preference-free and can be computed by taking the expectation of the option’s discounted payoff under an equivalent martingale measure (also called risk-neutral measure). The latter is a probability measure equivalent to (i.e. with exactly the same null-sets as) the real-world probability measure and under which asset prices and associated wealth processes discounted at the bank account’s interest rate are martingales, i.e. at each time, the present value is the best prediction for future values given past information. In other words under an equivalent martingale measure, risky assets have zero excess returns, i.e. same mean-return as with the riskless asset, e.g. as in the Black-Scholes model. The connection between risk-neutral pricing and martingale theory was first put into rigorous mathematical perspective by [HK79, HP81]. They characterized absence of arbitrage (also called free lunch) in a market with discrete-time trading by the existence of an equivalent martingale measure, their result is now known as the fundamental theorem of asset pricing. A consequence of this theorem and the classical predictable representation property of martingales is that completeness of the financial market is equivalent to uniqueness of the equivalent martingale measure. These results were later generalized to continuous-time trading by [DS94, DS98], in the context of asset prices being semimartingales. The latter publications introduced a reasonably general notion of market viability, namely the no-free lunch with vanishing risk (abbreviated NFLVR) condition, and linked it rather to a martingale (resp. local martingale, sigma-martingale) property of bounded (resp. locally bounded, unbounded) asset prices.

Page 6 In this thesis, we concentrate on pricing and valuation in the context of incomplete market where indeed many financial claims carry some inevitable risk. This may include, for instance, claims that are contingent on some non-tradeable underlying assets, e.g. weather derivatives or some volatility derivatives. Transaction costs, jumps in the underlying asset price, or unpredictable events in the information are possible reasons for incompleteness of a financial market. Convexity of the set of equivalent martingale measures implies that incomplete arbitrage-free markets admit infinitely many pricing measures; yielding notably an interval of risk-neutral prices for non-replicable claims. The issuer of a financial contract striving for robustness with respect to price misspecification and preference-freeness may therefore wish to sell at the upper bound over all possible no-arbitrage prices, so-called upper no-arbitrage bound. The buyer’s ideal valuation can be interpreted analogously as the corresponding lower bound. This valuation approach in incomplete markets was rigorously introduced by [EQ95], who showed that the upper no-arbitrage bound is exactly the minimal capital that allows the seller to super-replicate the claim in almost every state of the world by dynamically trading in the risky assets. The cost of super-replication being given by the supremum over all the risk-neutral prices, the corresponding wealth process was shown to be a supermartingale under any equivalent martingale measures and the associated super-replicating strategy can be derived via the optional decomposition theorem (see [Kra96, FK97, FK98]). Super-replication indeed is an extremely safe concept of hedging, since it excludes the possibility of losses and, while eventually allowing even for intermediate consumption, still ensures the terminal wealth to dominate the liability of the investor at delivery. Unsurprisingly, super-replication is often too costly and may not be appropriate for some practical applications; the minimal capital it requires for instance may be too high to find a buyer. For a long time, research in mathematical finance has been investigating about alternative approaches to (partial) hedging that require lower capital than super-replication would, hence making it more likely to find a buyer. Apart from the no-good-deal approach whose background we describe in the next paragraph and which lies at the heart of Chapter 2, 3 and 4, various solutions have been suggested in this regard: The quantile hedging approach of [FL99] allows the seller of a claim to charge a smaller amount to the buyer but still be able to dominate his liability with some target confidence level. This is more a hedging approach than a pricing one, since its main objective is to limit the risk of loss for the seller to a maximal pre-specified level by requiring a minimal capital that will make her position acceptable in this sense. In the same direction, one can consider risk minimization and mean-variance hedging whose objective are the minimization of a quadratic functional of the tracking (hedging) error of trading strategies. In order to achieve this for risk-minimization, one relaxes the self-financing requirement of the replicating strategy (corresponding to vanishing tracking error) and instead requires a notion of mean-self-financing strategy that corresponds to a martingale property of the tracking error. Risk minimization was first introduced by [FS86] in the situation where asset prices are modelled directly as martingales, and later extended in [FS90] to the general semimartingale

Page 7 case where it could only be defined in a local sense (local risk-minimization). The local riskminimizing strategy was ultimately derived via the so-called F¨ollmer-Schweizer decomposition of the wealth process, which can be viewed as a generalization to the semimartingale case of the well-known Galtchouk-Kunita-Watanabe decomposition from martingale theory. Naturally, a pricing concept is attached to risk-minimization via the so-called minimal martingale measure. However hedging according to some quadratic criterion has been criticized, mainly because it penalizes gains and losses in the same way. We will provide (cf. Chapter 3) a potential argument against this criticism by showing that if drift uncertainty is sufficiently large in the market, then risk-minimization coincides with robust good-deal hedging (in a suitable sense); the latter is using a non-quadratic hedging criterion by minimizing a dynamic coherent risk measure. If one rather insists on the self-financing property of the hedging strategy, then a quadratic hedging criterion leads to mean-variance hedging as studied in [BL89, DR91]. Also here one obtains a valuation that is consistent with the hedging criterion and can be computed under the variance-optimal martingale measure. A comprehensive survey of both approaches can be found in [Sch01]. However it turns out that the minimal and variance-optimal martingale measures may be only signed measures in general, and hence may lead to negative prices for some positive claims. This is clearly an undesirable feature of these two valuation approaches. However, for a more specific Markovian (incomplete) model of the stock price, for instance the Heston stochastic volatility model [Hes93], the minimal martingale measure can be written explicitly in terms of the market price of risk. In the Heston model the squared volatility process is a Cox-Ingersoll-Ross (CIR) process and the price of European call/put options under the minimal martingale measure is given by the Heston formula which is explicit up to the computation of a one dimensional improper integral. Using a single risk-neutral measure for valuation is clearly not conservative, as it introduces mark-to-market risk that can accumulate due to the necessary regular calibration. In Chapter 3, we suggest a more conservative approach in an example that shows how a robust valuation (and hedging) of volatility risk over a family of risk-neutral measures in the Heston model can be obtained, by restricting the mean-reversion level of the variance process to be within some confidence interval. Let us mention here also that a further alternative solution to the limitations of the abovementioned approaches is to take into account some utility-related preference of the investor or her aversion towards risk; this leads to rational pricing and hedging concepts that are consistent with the maximal expected utility of the investor. The literature in this direction is quite developed, and generally distinguishes between two approaches: the utility-indifference approach (cf. [HH09] for an overview and further references), and the utility-based approach (cf. [Dav97, HK04]). We do not say more about these approaches. Instead, we will present in Chapter 1 an example, for illustration of our JBDSE theory therein, that will deal with the solution to the classical expected utility maximization problem in incomplete market

Page 8 with additional liability. Note that this approach, somewhat problematically, assumes precise knowledge of the objective real-world probabilities. This is restrictive since model uncertainty, in particular about (highly uncertain) drift and volatility parameters under the real-world measure, is a problem in itself for practical applications. Good-deal valuation and hedging in the presence of model uncertainty will be studied in Chapters 3 and 4. In this thesis, we are mainly interested with the so-called no-good-deal approach to valuation of contingent claims in incomplete markets; cf. Chapter 2 to 4. As mentioned before, recall that no-arbitrage bounds are typically too large for most practical applications involving nonreplicable claims. The no-good-deal approach is a fairly conservative one that lies between using a single measure for pricing and using all equivalent martingale measures. Indeed the main idea is to obtain tighter valuation bounds, called good-deal bounds, by using as pricing measures only a subset of the equivalent martingale measures preventing some economically meaningful notion of good deals. The latter could be interpreted as trading opportunities that are too favorable and therefore should also be excluded from the market. Inherent to the concept of good-deal valuation is therefore already a certain notion of robustness (namely with respect to the smaller set of pricing measures as generalized scenarios). Good-deal bounds were introduced by [CR00] mostly in discrete time, interpreting good-deals as trading opportunities that admit an instantaneous mean excess return per unit volatility risk (called instantaneous Sharpe ratios) above a certain threshold. Their no-good-deal constraint therefore was imposed as a bound on the instantaneous Sharpe ratios in a financial market that is extended by additional price processes for derivatives. Their results were rigorously extended to continuous time by [BS06] in a Markovian model of asset prices and additional factor processes possibly exhibiting jumps. Using the so-called Hansen-Jagannathan (HJ) bounds (see [HJ91]), both papers showed that the constraint on the instantaneous Sharpe ratios can be obtained by pricing only under equivalent martingale measures satisfying a bound on the norm of their Girsanov kernels. The HJ bounds basically show that the maximal Sharpe ratio over all portfolio strategies cannot exceed the ratio of the standard deviation of a stochastic discount factor (i.e. the Radon-Nikodym derivative of the pricing measures) to its mean. In continuous (Brownian) filtrations, imposing a bound on instantaneous Sharpe ratios is basically equivalent to imposing a bound on the optimal expected growth rates [Bec09]. Such local no-good-deal constraints for pricing measures are favorable for good time-consistency properties of the resulting good-deal bounds; cf. [KS07b]. Following this remark, we will first consider in Chapters 2 and 3 a general theory of good-deal valuation and hedging for local constraints on Girsanov kernels given in terms of abstract correspondences. This will provide some flexibility as far as the choice of the no-good-deal constraint is concerned (e.g. in the jump setting of Chapter 2 where the Sharpe ratio constraint is no longer equivalent to the optimal growth rate one), but also will prove necessary in the presence of uncertainty, cf. Chapter 3, where the aggregate no-good-deal constraint under uncertainty may be different from any classical

Page 9 one. Note that good-deal bounds have also been defined by some notion of expected utilities [CH02, Cer03, KS07b]. Good-deal theory has been developed for a long time as a pure valuation approach (see [BY08, BL09, MMM13, Mur13] in a Brownian setting and [BS06, KS07b, Don11] in a setting with jumps). Contributions about hedging only appeared recently, mostly in the setting of a Brownian filtration. These started from [Bec09] who uses classical BSDEs to derive hedging strategies as minimizers of dynamic coherent risk measures [ADE+ 07] of no-good-deal type yielding the good-deal bound as the minimal capital for acceptability, i.e. the market consistent risk measure in the spirit of [BE09]. [CT14] studied mean-variance hedging in the context of good-deal valuations and concluded that both hedging approaches perform reasonably well. Throughout this thesis, we follow the good-deal hedging approach of [Bec09], for which valuations and hedges will be described by different classes of BSDEs (classical BSDEs, JBSDEs, 2BSDEs), depending on the framework in use. We note that hedging by minimizing a certain risk measure that allows for market consistent valuation is by now standard in the literature [CGM01, BE05, KS07a, BE09]. In addition, dynamic risk measures in general are well-connected to BSDEs; cf. [Ros06, PR15].

Robustness and model uncertainty in finance Robustness and model uncertainty are important topics in finance and decision theory; cf. [Con06, HS01]. Since definitions of the no-good-deal constraints involve the objective real-world probability measure, model uncertainty is also relevant to good-deal theory. Chapters 3 and 4 are concerned with robust approaches to uncertainty, in the Knightian sense (cf. [Kni21]), about the objective probability measure with respect to which good deals are defined, and good-deal bounds and hedging strategies are computed. In economic theory, it has been argued that incorporating uncertainty aversion provides a theoretical ground for explaining some behavioral observations such as the famous Ellsberg paradox [Ell61] or the equity premium puzzle [MP85]. Uncertainty in financial markets is a serious concern for (typically ambiguity-averse) investors who permanently strive for robustness in the valuation and hedging of their financial risks. Diverse mechanisms have been elaborated in the mathematical finance literature to take into account aversion towards uncertainty in financial modeling. In this thesis, we use a multiple prior approach to robustness under uncertainty proposed by [GS89, CE02], where an uncertaintyaverse investor or decision-maker seeks to protect herself against an eventual misspecification of probabilities by considering the most conservative (worst-case) line of action with respect to some confidence region of subjective probability measures called priors. The mathematical finance literature in this direction is wide and essentially distinguishes between drift uncertainty and volatility uncertainty. Following the same distinction, we will first consider drift uncertainty in Chapter 3 and then volatility uncertainty in Chapter 4, both in the context of good-deal

Page 10 theory. Drift uncertainty englobes in particular uncertainty about the market price of risk of traded assets, which naturally embeds into a setup where priors are equivalent to each other, i.e. they share the same nullsets and therefore agree about the impossible events in the market. This framework has been considered for instance in [DW92, Que04, GUW07, Sch08] for solving the maxmin expected utility maximization problem. In Chapter 3 we study robustness of good-deal hedging strategies for a worst-case approach to good-deal valuation, which yields larger good-deal bounds under uncertainty. We show the existence of a worst-case prior under which dynamic valuations and hedges can be computed as in the absence of uncertainty. For our results under drift uncertainty, we rely on classical BSDE methods under a fixed reference prior to which all others are equivalent. In the case of volatility uncertainty in Chapter 4, priors may no longer be equivalent to each other and we use 2BSDEs instead, for deriving valuation and hedging results. Historically, [ALP95, Lyo95] introduced the uncertain volatility model as a model of stock prices in the presence of volatility uncertainty, in which pricing and hedging of contingent claims in incomplete markets can be done in an analog way as in the (complete) Black-Scholes model. Typically, priors in the uncertain volatility model are mutually singular, since they may have disjoint supports; see e.g. [DM06, EJ13, EJ14]. This model is by now standard in the literature, and consists in modeling asset price dynamics under risk-neutral measures on the path-space that, being viewed as subjective priors, are parametrized by different volatility processes taking values in a pre-specified confidence interval of volatility values. [ALP95, Lyo95] (see also [Vor14] for a model of stock prices as geometric G-Brownian motions) derived no-arbitrage valuation bounds for financial derivatives in terms of the solution to a fully nonlinear PDE called the Black-Scholes-Barenblatt equation, which is a nonlinear analog of the Black-Scholes PDE in the presence of volatility uncertainty. In particular for convex payoff functions they showed that the worst-case model for the upper valuation bound corresponds to the highest volatility under which the two pricing PDEs coincide. In general when priors are non-dominated (i.e. they may disagree about the impossible market scenarios), one has to resort to different techniques for dynamic formulations and solutions of robust stochastic optimization problems; see e.g. [EJ14]. A typical difficulty in this case appears when defining the essential supremum of a family of random variables, which in the dominated case is well-defined up to a null set for a dominating prior. However if the priors are mutually singular, the definition of essential supremums necessitates some aggregation procedures for the null-sets of priors, which can then be disjoint. The quasi-sure analysis of [DM06]) provides a suitable framework for dealing with these technical issues, and is used for example in [DK13a, MPZ15] for maxmin expected utility maximization under volatility uncertainty. For our 2BSDE approach in Chapter 4, the quasi-sure analysis will be used naturally following the wellposedness theory of Lipschitz 2BSDEs in [STZ12]. In this chapter, robustness of good-deal hedging strategies for worst-case good-deal

Page 11 valuation under volatility uncertainty will be shown. Due to the technical issues mentioned above, we are not able to show existence of a worst-case prior for dynamic good-deal bounds in the general theory. However, in an example for European put options in a (two dimensional) Black-Scholes model for a traded and a non-traded asset, we will constructively identify a worst-case prior for dynamic valuations which mimics the relation to convexity of the payoff function as in [ALP95, Lyo95, Vor14]. A closed-form expression of the robust hedging strategy will be subsequently given, after explicitly identifying the solution to the 2BSDE in the example. Let us mention that recent developments in robust finance include the drift-and-volatility uncertainty framework of [EJ13, EJ14] for formulation of the pricing, hedging and maxmin expected utility maximization problems in a continuous dynamic setting, taking into account the investor’s uncertainty about both volatility and drift. Solutions to the robust utility maximization problem under drift-and-volatility uncertainty in continuous time have been investigated recently by [BP15] using PDE methods, focusing on ellipsoidal drift-uncertainty for each fixed volatility scenario. Although dealing only with drift uncertainty, Chapter 3 will consider some cases where the confidence set of drift uncertainty is also described by an ellipsoid, which seems natural for drift uncertainty modeling; cf. also [GUW07]. Note that [Nut14] also recently showed existence of an optimal trading strategy for the maxmin utility-maximization problem, for arbitrary sets of priors and bounded utility functions, but restricting his analysis to discrete time. Drift-and-volatility uncertainty is however not the route followed in this thesis, as both types of uncertainty will be considered separately. This is partly motivated by the fact that in our dynamic setting standard conditions for wellposeness of 2BSDEs (in particular regularity and convexity of the generator) as in [STZ12] may not hold for the dynamic good-deal valuation and hedging problem in Chapter 4 if one considers drift uncertainty in addition.

Contribution of the thesis This thesis is organized in four chapters which are mostly self-contained and can be read almost independently. The connections between the chapters’ results can be specified as follows: Chapter 1 deals with a theoretical study of wellposedness and comparison for solutions to BSDEs with jumps of infinite activity and time-inhomogeneous compensators; Chapters 2 to 4 are concerned with good-deal valuation and hedging. More precisely, Chapter 2 applies some results of Chapter 1 to market models that allow for jumps described by abstract random measures; Chapters 3 and 4 finally study robustness (of valuation and hedges) with respect to uncertainty about the drift of traded assets for the former and about their volatility for the latter. A more detailed chapter-wise description of the contribution of the thesis will now be given below.

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Chapter 1: Concrete criteria for wellposedness and comparison of BSDEs with jumps of infinite activity This chapter is based on [BBK15] and studies bounded solutions (Y, Z, U ) to JBSDEs. Recall that in comparison to classical BSDEs which are driven only by a Brownian motion W , such JBSDEs are additionally driven by a random measure µ ˜ and involve a second stochastic integral with respect to the compensated random measure µ ˜ = µ − ν P with integrand U on which the generator f (Y− , Z, U ) may also depend. We extend the analysis of JBSDEs beyond classical Lipschitz assumptions (cf. [TL94, BBP97]) on the generator f by concentrating on a family of generators that satisfy a certain monotonicity property, but do not need to be globally Lipschitz in the U -component; see also [Bec06, Roy06]. We do not require the compensator ν(dt, de) of µ(dt, de) to be a product measure like λ(de)⊗dt, as it would be natural for random measures of jumps of L´evy type for instance. Instead, ν is only assumed to be absolutely continuous to some reference product measure λ ⊗ dt with a bounded Radon-Nikodym derivative ζ (implying time-inhomogeneity) where λ is a σ-finite measure, hence allowing for infinite activity of the driving jumps. This provides wide scope for stochastic dependencies between W and µ ˜, which can be relevant in applications; cf. examples in [BS05]. Furthermore, it embeds a range of interesting driving processes for BSDEs in addition to Brownian motion, including L´evy processes, Poisson random measures, marked point processes, Markov chains or much more general step processes (as in [HWY92], Chapter 11, including e.g. semi-Markov processes), connecting our analysis to research from [NS01, CE10]. As usual in the BSDE literature, we require a key property on the filtration, namely that µ ˜ and W together have the weak predictable representation property for martingales. In order to deal with the time-inhomogeneous setting, we slightly extend a general but technical comparison theorem on JBSDE from [Roy06] by using a more general (Aγ )-condition, in order to derive sufficient conditions for comparison which are easier to verify, since they are formulated in terms of concrete properties of generator functions from our family of interest. This gives rise to a-priori estimates of the L∞ -norm of the Y -component of the JBSDE solution. Additionally we obtain existence and uniqueness results for bounded JBSDE solutions (i.e. wellposedness of such JBSDEs) in the case of jumps with finite activity, as in [Bec06]. These steps enable us to advance to the case of infinite activity. To this end, we first approximate the generator by truncating the activity of the jumps using the σ-finiteness of λ. This leads to a monotone sequence of generators for which solutions do uniquely exist. Then using monotone stability arguments like in [Kob00] enables us to obtain wellposedness for the initial JBSDE generator. However, it turns out that such arguments only work at first for terminal conditions ξ which are small in L∞ -norm. By pasting solutions for sufficiently small terminal conditions one can show convergence to the bounded solution of the JBSDE for the original data (ξ, f ). For this purpose, we follow the iterative idea of [Mor10] who focused on a particular generator,

Page 13 and elaborate the proof slightly differently and more compactly for our general setting. For results in related but different directions, we mention [CM08] for comparison of JBSDEs with (doubly) reflection for Lipschitz generators, [KMPZ10] for (minimal) solutions with constraints on jumps for Poisson random measures of finite activity, [DI10] for time delayed generators and [KTPZ15a] for a pasting argument for quadratic JBSDEs following the fixed-point approach of [Tev08]. We note that our results are mainly stated for generators that are Lipschitz continuous in the Zcomponent. For results on generators of quadratic growth in Z we refer e.g. to [Mor10, EMN14, KTPZ15a]. [EMN14] work in finite activity and [Mor10] considers a particular generator function arising from a specific utility maximization problem. As in our setup [KTPZ15a] also works in infinite activity and considers time-inhomogeneous compensators. However, the applicability of their results may be less straightforward than ours due the abstract nature of their assumptions on the generator which are stated in terms of existence of some abstract processes satisfying strong integrability requirements such that certain non-linear estimates hold without exception of a null set. Another contribution of this chapter is therefore that we provide concrete conditions on generators that are easier to verify in applications. For illustration purposes, we apply our results to the utility maximization problem in finance, for power and exponential utility functions complementing results e.g. from [HIM05, Sek06, Bec06, Mor10, Nut12a]. In Chapter 2, a nonlinear example on good-deal valuation and hedging in incomplete markets with jumps will be also covered.

Chapter 2: Hedging under generalized good-deal bounds in jump models with random measures In this chapter, which is based on [BK15c], we study good-deal hedging and valuation in general jump models driven by random measures. More precisely, we suppose the presence of unpredictable event-risk in the market in the sense that the information flow (filtration) may be discontinuous, allowing for non-trivial purely-discontinuous price processes with totally inaccessible jump times. [BS06] first considered good-deal valuation in a Markovian setting with jumps, where good-deal bounds were defined by a constraint on the instantaneous Sharpe ratios and derived subsequently as solution to HJB equations. Although [BS06] focused only on valuation, they raised the crucial need for a hedging theory that corresponds to good-deal valuation. A first attempt to good-deal hedging was by [Bec09] in a Brownian setting, who derived hedging strategies from minimizing suitable dynamic coherent risk measures that allow for optimal risk sharing with the market through good-deal valuation. Another attempt, still in a Brownian setting, is based on a quadratic hedging criterion like mean-variance hedging and was developed by [CT14]. In the jump setting, [Del12] studied the approaches of [Bec09, CT14] for point-processes with state-independent jump intensities, restricting to a market with two

Page 14 risky assets (a traded and a non-traded one) and considering solely Sharpe ratio no-good-deal constraints. For a related problem, we study in this chapter a generalization to multiple non-necessarily Markovian risky assets, more general jump processes driven by abstract integervalued random measures and generalized no-good-deal constraints on Girsanov kernels of pricing measures parametrized by correspondences (i.e. set-valued mappings as studied in [AF90]). Using correspondences provides an abstract framework for incorporating no-good-deal restrictions of different natures (e.g. instantaneous Sharpe ratios, optimal growth rates, instantaneous Sharpe ratios under uncertainty about jump intensities, etc.). We derive generalized gooddeal valuation bounds for possibly path-dependent contingent claims using classical JBSDEs instead of HJB equations (under Markovian assumptions) as in [BS06, Don11, Mur13]. We first consider the case of uniformly bounded correspondences, for which the generators of the resulting JBSDEs are Lipschitz continuous. The resulting valuation JBSDE is derived from the comparison principle of Chapter 1, and its generator is the maximum of a family of linear JBSDE generators parametrized by the Girsanov kernels of no-good-deal pricing measures. This generator in general does not have an explicit form, even for the classical radial (Sharpe ratio) no-good-deal constraint. To obtain more constructive (or even explicit) expressions of the generators, we assume more structure on the contingent claim, the nature of the no-good-deal constraint or the random measure of the jumps. Examples are presented with closed-form expressions for the associated good-deal hedging strategy. The case beyond uniform boundedness of the correspondence is also considered, where the Lipschitz property of the generators is no longer ensured and approximation arguments are used. Using a notion of good-deal hedging introduced by [Bec09], we contribute results on the existence of hedging strategies for arbitrary bounded correspondences. Moreover we obtain a characterization of the hedging strategies in terms of the solution to the JBSDE describing the good-deal valuation bound. We show that the tracking errors of hedging strategies, i.e. the dynamic difference between the good-deal bound and the profit/loss from trading, satisfy a supermartingale property under some a-priori valuation measures including the no-good-deal measures. As the martingale property of the tracking errors corresponds to hedging strategies being mean-self-financing in the terminology of [Sch01], this means that the good-deal hedging strategies can be viewed as being at least mean-self-financing under every a-priori valuation measure. The latter can be interpreted as a robustness property of the good-deal hedging strategy with respect to the set of a-priori valuation measures as generalized scenarios in the sense of [ADE+ 07]. For concrete no-good-deal constraints we provide some examples where a good-deal hedging strategy can be obtained explicitly in terms of solutions to JBSDEs. Hedging is investigated only for bounded correspondences, and in the case beyond uniform boundedness we only present results about good-deal valuations. In a discontinuous filtration, imposing a bound on the instantaneous Sharpe ratios via a bound on the norms of the Girsanov kernels of pricing measures (as in [BS06]) is not equivalent to

Page 15 imposing a bound on the optimal conditional expected growth rates (as in [Bec09]). We note that the growth rate constraint is mathematically less tractable than the Sharpe ratio one, at least in terms of using Lipschitz BSDEs. Indeed, it turns out that Sharpe ratio constraints fit well with the theory of Lipschitz JBSDEs for arbitrary random measures. For such constraints and for some concrete random measures of jumps, we can even obtain more simplified JBSDE descriptions of good-deal bounds and hedging strategies. In particular for jumps of a continuoustime Markov chain and without a Gaussian component, we infer from [CS12] that the JBSDE for good-deal bounds defined from Sharpe ratio constraints, for Markovian European contingent claims depending only on the terminal value of the chain, reduces to a fully-coupled system of ordinary differential equations (ODEs). The latter can be transformed (by reversing time ) into an initial value problem, which can then be solved using any standard numerical ODE solver. For Sharpe ratio constraints, we also present an example for robust hedging under Knightian uncertainty about the intensity of the underlying jump process, linking the result here with those of Chapters 3 and 4. Here robustness of a hedging strategy with respect to uncertainty refers to a property of being at least mean-self-financing under every a-priori valuation measure, uniformly with respect to a family of subjective probability measures as candidates for the real-world measure and capturing the uncertainty. On the other hand, optimal growth rate constraints do not fit well with Lipschitz BSDEs since the resulting correspondence may not be uniformly bounded. For such constraints it turns out that we can still rely on the theory of Lipschitz JBSDEs when random measures with finite support of the compensator. Results are then obtained for finite-state semi-Markov processes, which are a flexible class with many practical applications, see e.g. for actuarial applications [BMS14] and references therein.

Chapter 3: Hedging under generalized good-deal bounds and drift uncertainty This chapter is based on [BK15b] and is concerned with approaches to good-deal hedging (as in [Bec09]) under ambiguity about the objective probability with respect to which good deals are defined and good-deal bounds computed. Good-deal valuations fit into the theory of dynamic monetary convex risk measures (or monetary convex utility functionals) for which results, in particular about dual representations and time consistency, exist in high generality; see e.g. [KS07a, BNN13, DK13b]. We contribute constructive and qualitative results on the (robust, good-deal) hedging strategies, that facilitate interpretation and are accessible to computation. We pose the good-deal valuation and hedging problem in a framework with multiple priors, and follow a robust worst-case approach as in [GS89, CE02]. We note that results on good-deal valuations and hedges under uncertainty in a recent work by [BCCH14] are very different to ours. Indeed, they work mostly in discrete time and study numerical results for a different uncertainty-penalized preference functional, whereas we use dynamic coherent risk measures in continuous time and focus on rather analytical results.

Page 16 After formulating a framework with predictable correspondences as in Chapter 2 but in the Wiener setting, we also describe good-deal hedging strategies and valuation bounds in terms of solutions to classical BSDEs. In the absence of uncertainty, we obtain results first for uniformly bounded correspondences and then for the case beyond uniform boundedness by approximation. Additionally we characterize the good-deal valuation bounds in the possibly unbounded case in terms of minimal supersolutions to convex BSDEs (as in [DHK13]). Notably, the abstract generalized constraints are needed to cover relevant examples of uncertainty about the market prices of risk of the assets that are available in the (incomplete) market for dynamic partial hedging. For illustration purposes, we will consider e.g. ellipsoidal correspondences which permit explicit analytic generators in the BSDEs of interest, being efficient for Monte-Carlo approximation. In general good-deal hedging strategies can comprise a speculative bet in the direction of the market price of risk to compensate for unhedgeable risks. In this chapter we also provide new examples on good-deal valuation and hedging, with closedform formulas for good-deal bounds and hedging strategies: For an exchange option between tradeable and non-tradeable assets, we give a Margrabe-type formula [Mar78] for the good-deal bound, with adjusted input parameters. For the stochastic volatility model by Heston [Hes93], we obtain semi-explicit formulas under good-deal constraints for pricing measures, which restrict the mean-reversion level of the stochastic variance process to be within some confidence interval. A graphical analysis of the dependency on model parameters is also done. An interesting aspect of the latter example shows, how a robust valuation of volatility risk (over a family of no-good-deal pricing measures) can be obtained for an absolutely continuous family of measures. To illustrate to which extend our BSDE solutions could be computed by efficient but generic Monte-Carlo methods, complementing numerical approaches to hedging from [CT14], we investigate the errors between the Monte-Carlo approximations and our analytic formula in a four dimensional example for an exchange option. In the presence of uncertainty, we derive general results for good-deal bounds and hedging strategies that are robust with respect to uncertainty, described also by correspondences. Building on a suitable definition of good-deal bounds in the presence of uncertainty, we note that the problem with multiple priors can be related to a respective problem without uncertainty but with an enlarged good-deal constraint correspondence, which even in the most natural cases of no-good-deal restriction and uncertainty may easily not have a radial shape; hence the need of a general theory for abstract correspondences in the first place. The worst-case approach naturally leads to a robust notion of valuation by the widest good-deal bounds that are obtained over all probabilistic models under consideration. We show that there is also a notion for robust hedging, which corresponds to the aforementioned robust good-deal valuation. Indeed, there exists a unique strategy that is robustly at least mean-self-financing, in the sense that it is at least mean-self-financing (see Chapter 2) uniformly with respect to all priors. By saddle point arguments we derive a minmax identity, that shows how the robust good-deal hedging strategy

Page 17 is given by the (ordinary) good-deal hedging strategy with respect to a worst-case measure. Since we rely on BSDEs both are actually identified in a constructive manner. As intuition suggests, a robust approach to uncertainty reduces the speculative component of the good-deal hedging strategy. As a further contribution, we prove that if the uncertainty is large enough in relation to the no-good-deal constraints, then the robust good-deal hedging strategy does no longer include any speculative component, but coincides with the (globally) risk-minimizing strategy of [FS86]. This offers theoretical support to the commonly held perception that hedging should abstain from speculative objectives (see e.g. [LP00]), and moreover a new justification for risk-minimization. Finally, an example with closed-form solutions for robust good-deal bounds and hedging strategies for an option on a non-traded asset illustrates results and graphically analyzes dependencies on parameters. This chapter has built over the Masters’ thesis [Ken11] for preliminary results including those about generalized good-deal bounds in the absence of uncertainty for uniformly bounded and ellipsoidal correspondences, and part of worst-case valuation in the presence of uncertainty. All remaining results are new; among others, the examples with closed-form expressions for the good-deal bound and hedging strategy, the saddle-point results on worst-case valuation and hedging in the presence of uncertainty, and the link to risk-minimization obtained in the last section.

Chapter 4: Hedging under good-deal bounds and volatility uncertainty: a 2BSDE approach Chapter 4 is based on [BK15a] and deals with robust good-deal hedging and valuation with respect to volatility uncertainty. We consider also here an approach under which good-deal bounds are computed as worst-case valuations over a calibrated class of priors. Contrary to drift uncertainty for which BSDE descriptions can be given under a single dominating prior (see Chapter 3), volatility uncertainty corresponds to priors that are mutually singular, and therefore necessitates a different mathematical framework for valuation and hedging results. In particular a rigorous definition of the dynamic good-deal bound as essential supremum/infimum of random variables involves additional technical care since it may not be possible to aggregate the null-sets of the different priors. First we present some purely theoretical results on the comparison principle for solutions to 2BSDEs with different generators and terminal conditions, thus extending a result in [STZ12] for which the generators are identical. We use the so-called strong formulation of volatility uncertainty, which considers the uncertain priors as local martingale laws of stock price processes defined on the canonical Wiener space. Our definitions of worst-case good-deal bounds and hedging strategies are adapted to this framework and we follow a setup by [STZ13], which starting from the canonical space and working with regular conditional probability distributions

Page 18 (in short r.c.p.d.) ensures a time-consistency property (in the spirit of [NS12]) of the good-deal bounds as dynamic risk measures under volatility uncertainty. This paves the way for defining good-deal hedging again as minimization of residual risk from dynamic trading. In this chapter, we concentrate on a no-good-deal restriction imposed as a bound on the instantaneous Sharpe ratios under each reference prior separately. This provides a family of good-deal bound processes parametrized by priors, and the worst-case upper good-deal bound arises as the largest among them, i.e. their essential supremum. Building on the intuition from Chapter 3, we derive robust good-deal bounds and hedging strategies under volatility uncertainty in terms of solutions to Lipschitz 2BSDEs relying on the theory in [STZ12]. Again as in Chapter 3 robustness of the good-deal hedging strategy with respect to uncertainty is related to the property of being at least mean-self-financing uniformly over all priors. Finally, we contribute an example for put options on non-traded assets under volatility uncertainty, in which a worst-case model can be explicitly computed for the dynamic good-deal bound and closed-form formulas for robust valuations and hedges are derived, like in [ALP95, Lyo95, Vor14]. The latter works focus on robust superhedging in the presence of volatility uncertainty, whereas we focus on good-deal hedging. As an example demonstrates, our robust good-deal hedging strategy (and respective valuation bounds) can in general be very different from the super-replicating one.

1. Concrete criteria for wellposedness and comparison of BSDEs with jumps of infinite activity In this chapter, we study JBSDEs for a specific class of generator functions that do not necessarily satisfy global Lipschitz conditions in the jump integrand. The JBSDEs in consideration are driven, additionally to a Brownian motion, by general random measures with compensators that can be inhomogeneous in time and may allow for infinite activity of the jumps of the value process. In this context, we provide in Sections 1.2 and 1.3 concrete conditions that are directly verifiable for existence, uniqueness and comparison of bounded solutions to such JBSDEs, first in the case of finite activity of the jumps and then to the infinite activity case by suitable approximations. Section 1.4 illustrates the range of applicability of our results by solving the utility maximization problem in finance for exponential and power utility functions with additive and multiplicative liability respectively. To make this chapter as self-contained as possible, we first introduce some useful notations and the mathematical preliminaries.

1.1

Mathematical framework and preliminaries

This section presents the technical framework and sets the notations. We will also summarize the key assumptions on the BSDE generator (1.7) which will play, in varying combinations, a role in our later results. First we recall essential facts on stochastic integration with respect to random measures and on bounded solutions for Backward SDEs which are driven jointly by Brownian motions and a compensated random measure. For notions from stochastic analysis not explained here we refer to [JS03] and [HWY92]. Inequalities between measurable functions are understood almost everywhere with respect to an appropriate reference measure, typically P or P ⊗ dt. Let T < ∞ be a finite time horizon and (Ω, F, (Ft )0≤t≤T , P ) a filtered probability space with a filtration satisfying the usual conditions of right continuity and completeness, assuming FT = F and F0 being trivial (under P ). Due to the usual conditions we can and do take all semimartingales to have right continuous paths with left limits, so-called c`adl`ag paths. Expectations under a probability Q and Conditional expectations given Ft are denoted by E Q [·] and EtQ [·] respectively, or simply E[·] and Et [·] when Q = P . Reference to the probability is omitted if clear from context. Let H be a separable Hilbert space and we denote by B(E) the Borel σ-field of E := H\{0}, e.g. H = Rl , l ∈ N or H = `2 ⊂ RN . Then (E, B(E)) is a standard Borel space. In addition, let W be a d-dimensional Brownian motion. Stochastic integrals of a vector valued predictable process Z 19

Section 1.1. Mathematical framework and preliminaries

Page 20

with respect to a semimartingale X, e.g. X = W , of the same dimensionality are scalar valued R R semimartingales starting at zero and denoted by (0,t] ZdX = 0t ZdX = Z · Xt for t ∈ [0, T ]. The predictable σ-field on Ω × [0, T ] generated by all left continuous adapted processes is e := Ω × [0, T ] × E. denoted by P and Pe := P ⊗ B(E) is the respective σ-field on Ω Let µ be an integer-valued random measure with compensator ν = ν P (under P ) which is assumed to be absolutely continuous to λ ⊗ dt for a σ-finite measure λ on (E, B(E)) satisfying R 2 e E 1 ∧ |e| λ(de) < ∞ with some P-measurable, bounded and non-negative density ζ, such that ν(dt, de) = ζt (e) λ(de) dt = ζt dλ dt, (1.1) with 0 ≤ ζt (e) ≤ cν P ⊗ λ ⊗ dt-a.e. for some constant cν > 0. L2 (λ) (resp. L2 (ζt dλ)) R defines the space that of E-measurable functions γ : E → R with E |γ(e)|2 λ(de) < ∞ (resp. R 2 2 2 E |γ(e)| ζt (e)λ(de) < ∞). Note that the Hilbert spaces L (λ) and L (ζt dλ), are separable since the underlying measures are σ-finite and the σ-algebra E is countably generated (see [Coh13, Proposition 3.4.5]). Hence they admit countable orthonormal bases and are in particular Polish spaces. Since the density ζ can depend on t and ω, the compensating measure ν may be time-inhomogeneous and stochastic. This permits for a richer dependence structure between W and µ; for instance the density ζ and thereby the intensity of the jump measure might fluctuate in dependence of some diffusion process driven by W . Let Q be a probability measures. We denote by Lp (Q), 1 ≤ p < ∞, the space of FT measurable random variable X with kXkpLp (Q) := E Q [|X|p ] < ∞, and L∞ (Q) the space of FT -measurable random variable kXkL∞ := kXk∞ = ess supQ |X| < ∞. For a function U : [0, T ] × Ω × E → R we define |U |∞ := ess sup(t,e) |Ut (e)|. For stochastic integration with e and W we define sets of R-valued processes respect to µ ( p

S (Q) :=

!

Y c`adl`ag : |Y |p := E (



S (Q) :=

Y c`adl`ag : |Y |∞

Q

p

sup |Yt |

:=

e U P-measurable :

for p ∈ [1, ∞) ,

−1 and E(exp(hγ ∗ µ eiT )) = E exp 0T E |γ s (e)|2 ν(ds, de) 1. ∆(γ ∗ µ < ∞ (see R 2 Theorem 9, [PS08]). In particular, this holds if E |γ s (e)| ζs (e) λ(de) < const. < ∞ P ⊗ ds-a.e. and γ > −1. R

e) ≥ −1 + δ for some δ > 0 and γ ∗ µ e is a BM O(P )-martingale. This is due to 2. ∆(γ ∗ µ Kazamaki [Kaz79]. e) ≥ −1 and γ ∗ µ e is a uniformly integrable martingale and E(exp(hγ ∗ µ eiT )) < ∞ 3. ∆(γ ∗ µ (see Theorem I.8, [LM78]). Such a condition is satisfied when γ is bounded and |γ| ≤ ψ, P ⊗ dt ⊗ dλ-a.s. for a function ψ ∈ L2 (λ) and ζ ≡ 1. The latter is what is required for instance in the comparison Theorem 4.2 of [QS13]. R

e) for β Note that under above conditions, also the stochastic exponential E( βdW + γ ∗ µ bounded and predictable is a martingale, as it is easily seen by Novikov’s criterion.

In the statement of Proposition 1.4, the dependence of the process γ on the BSDE solutions is not needed for the proof as the same result holds if γ is just a predictable process such that the estimate on the generator f2 and the martingale property (1.9) hold. The further functional dependence is needed for the sequel, as required in the following Definition 1.7. An R-valued generator function f is said to satisfy condition (Aγ ) if there is a P ⊗ B(Rd+3 ) ⊗ B(E)-measurable function γ : Ω × [0, T ] × Rd+3 × E → (−1, ∞) given 0 by (ω, t, y, z, u, u0 , e) 7→ γty,z,u,u (e) such that for all (Y, Z, U, U 0 ) ∈ S ∞ × H2 × (Hν2 )2 with 0 |U |∞ < ∞, |U 0 |∞ < ∞ it holds for γ := γ Y− ,Z,U,U ft (Yt− , Zt , Ut )−ft (Yt− , Zt , Ut0 ) ≤

Z E

γ t (e)(Ut (e) − Ut0 (e))ζt (e)λ(de), P ⊗ dt-a.s.

(1.12)

R

e) is a martingale for every bounded and predictable β. and E( βdW + γ ∗ µ

A function f satisfies condition (A0γ ) if the above holds for all bounded U and U 0 with additional e ∈ BM O(P ) and U 0 ∗ µ e ∈ BM O(P ). property that U ∗ µ Clearly, existence and applicability of a suitable comparison result of solutions to JBSDEs implies their uniqueness. In other words assuming there exists a bounded solution for a Lipschitz driver with respect to y and z which satisfies (Aγ ) or (A0γ ), we obtain that such a solution is unique. 0

Example 1.8. A natural candidate γ for drivers f of the form (1.7) is given by γsy,z,u,u (e) := R1 ∂ 0 0 ∂u gs (y, z, tu + (1 − t)u , e) dt 1A (e), assuming differentiability of g. Indeed, we have 0

1

 ∂  (gs (y, z, tu + (1 − t)u0 , e)) dt 1A (e) 0 ∂t = (gs (y, z, u, e) − gs (y, z, u0 , e))1A (e),

γsy,z,u,u (e)(u − u0 ) =

Z

Section 1.2. Comparison theorems and a-priori-estimates

Page 28

and hence the function γ is P ⊗ B(Rd+3 ) ⊗ B(E)-measurable since y,z,u,u0

γs

  gs (y,z,u,e)−gs (y,z,u0 ,e) 1 (e), A u−u0 (e) = 0,

u 6= u0 u = u0 ,

∂g are P ⊗ B(Rd+3 ) ⊗ B(E)-measurable. By the mean value theorem γ has the form and g, ∂u 0

γsy,z,u,u (e) =

∂ g(s, y, z, v, e)1A (e), ∂u

(1.13)

for some v between u and u0 . For generators of the form (1.8) γ simplifies to 0 γsy,z,u,u (e)

Z

= 0

1

∂ gs (tu + (1 − t)u0 , e) dt 1A (e) . ∂u

Definition 1.9. A generator f satisfies condition (Af in ) (on D or for elements in D) or (Ainf i ) if 1. (Af in ): f is of the form (1.7) with λ(A) < ∞, is Lipschitz continuous with respect to y and z, and the map u 7→ g(t, y, z, u, e) is continuously differentiable for all (ω, t, y, z, e) (in D) such that the derivative is strictly greater than −1 (on D ⊆ Ω×[0, T ]×R×Rd ×E) and locally bounded (in u) from above, uniformly in (ω, t, y, z, e). 2. (Ainf i ): f is of the form (1.8), is Lipschitz continuous with respect to y and z, and the map u 7→ gt (u, e) is twice continuously differentiable for all (ω, t, e) with the derivatives being locally (in u) bounded uniformly in (ω, t, e), the first derivative bounded away from ∂g −1 with a lower bound −1 + δ for some δ > 0, and ∂u (t, 0, e) ≡ 0. Example 1.10. Sufficient conditions for (Aγ ) and (A0γ ) are 1. γ is a P ⊗ B(Rd+3 ) ⊗ B(E)-measurable satisfying the inequality in (1.12) and 0

C1 (1 ∧ |e|) ≤ γty,z,u,u (e) ≤ C2 (1 ∧ |e|), for some C1 ∈ (−1, 0] and C2 > 0 on E = Rd \{0}. In this case exp h βdW + γ ∗  R eiT is clearly bounded and the jumps of βdW + γ ∗ µ e are bigger than −1. Hence µ R e E ( βdW + γ ∗ µ) is a positive martingale ([PS08], Theorem 9). Thus Definition 1.7 generalizes the original (Aγ )-condition introduced in [Roy06] for Poisson random measures. R

2. (Af in ) is sufficient for (Aγ ). This follows from Example 1.6.1, (1.13) and λ(A) < ∞. 3. (Ainf i ) is sufficient for (A0γ ). To see this, let u, u0 be bounded by c and γ be the natural candidate in Example 1.8. By the mean value theorem there exist v(e) between u and u0

Section 1.2. Comparison theorems and a-priori-estimates

Page 29

and ve(e) between 0 and v(e) such that ∂ g(s, v(e), e)1A (e) ∂u   ∂ ∂ ∂2 = g(s, v(e), e) − g(s, 0, e) 1A (e) = v(e) 2 g(s, ve(e), e)1A (e). ∂u ∂u ∂u

0

γsy,z,u,u (e) =

0

So γsy,z,u,u (e) is bounded uniformly in (ω, s, e) by 2 y,z,u,u0  ≤ sup ∂ g(s, y, z, u, e) |u| + |u0 | , γ (e) s 2 |u|≤c

∂u

e is a BMO-martingale by the BMO-property of U ∗ µ e and U 0 ∗ µ e hence βdW + γ ∗ µ R e) is a martingale by with some lower bound −1 + δ for its jumps. And E( βdW + γ ∗ µ Kazamaki’s criterion of Example 1.6. R

As an application of the above, we can now provide simple conditions for comparison in terms of concrete properties of the generator function, which are much easier to verify than the more general but abstract conditions on the existence of a suitable function γ as in Proposition 1.4 or the general conditions by [CE10]. Note that no convexity is required in the z or u argument of the generator. The result will be applied later to prove existence and uniqueness of JBSDE solutions. Theorem 1.11 (Comparison Theorem). A comparison result between bounded BSDE solutions in the sense of Proposition 1.4 holds true in each of the following cases: 1. (finite activity) f2 satisfies (Af in ). e and U 2 ∗ µ e are BMO(P )-martingales 2. (infinite activity) f2 satisfies (Ainf i ) and U 1 ∗ µ 1 1 for the corresponding JBSDE solutions (Y , Z , U 1 ) and (Y 2 , Z 2 , U 2 ).

Proof. This follows directly from Proposition 1.4 and Example 1.10, noting that representation (1.13) in connection with condition (Af in ) resp. (Ainf i ) meets the sufficient conditions in Example 1.6. Unlike classical a-priori estimates that offer some L2 -norm estimates for the BSDE solution in terms of the data, the next result gives a simple L∞ -estimate for the Y -component of the solution. Such will be useful for the derivation of BSDE solution bounds and for truncation arguments. Proposition 1.12. Let (Y, Z, U ) ∈ S ∞ × H2 × Hν2 be a solution to the BSDE (ξ, f ) with ξ ∈ L∞ (FT ). Let f be Lipschitz continuous with respect to (y, z) with Lipschitz constant Kfy,z and satisfying (Aγ ) with f. (0, 0, 0) bounded. Then |Yt | ≤ exp Kfy,z (T − t) |ξ|∞ + (T − t)|f. (0, 0, 0)|∞ 



Section 1.2. Comparison theorems and a-priori-estimates

Page 30

holds for all t ≤ T . Proof. Set (Y 1 , Z 1 , U 1 ) = (Y, Z, U ), (ξ 1 , f 1 ) = (ξ, f ), (Y 2 , Z 2 , U 2 ) = (0, 0, 0) and (ξ 2 , f 2 ) = (0, f ), Then following the proof of Proposition 1.4, equation (1.11) becomes (RY )τ ∧t ≤ (RY )τ ∧T +

Z

τ ∧T

τ ∧t

Rs fs (0, 0, 0) ds − (LτT − Lτt ),

t ∈ [0, T ],

e for all stopping times τ where L := M − hM, N i is in Mloc (Q), M := RZ dW + (RU ) ∗ µ R 2 0,0,U,0 e with γ := γ is in M , N := β dW + γ ∗ µ and the probability measure Q ∼ P is given by dQ := E(N )T dP . Localizing (Lt )0≤t≤T along some sequence (τ n )n∈N ↑ ∞ yields R ∧T EtQ [(RY )τ n ∧t ] ≤ EtQ [(RY )τ n ∧T + ττ∧t Rs fs (0, 0, 0) ds]. By dominated convergence, we conclude that P -a.e R

"

Yt ≤

EtQ

RT ξ+ Rt

Z t

T

#

y,z Rs fs (0, 0, 0) ds ≤ eKf (T −t) (|ξ|∞ + (T − t)|f· (0, 0, 0)|∞ ). Rt

e with γ e := γ 0,0,0,U , and Q equivalent to P via Analogously, if we define N := β dW + γe ∗ µ dQ := E(N )T dP , we deduce that L := M − hM, N i ∈ Mloc (Q) and R

(RY )τ ∧t ≥ (RY )τ ∧T +

Z

τ ∧T

τ ∧t

τ

τ

Rs fs (0, 0, 0) ds − (LT − Lt ),

t ∈ [0, T ],

for all stopping times τ . This yields the lower bound. Again, we can specify explicit conditions on the generator function that are sufficient to ensure the more abstract assumptions of the previous result. Theorem 1.13. Let (Y, Z, U ) ∈ S ∞ × H2 × Hν2 be a solution to the BSDE (ξ, f ) with ξ in L∞ (FT ), f being Lipschitz continuous with respect to y and z with Lipschitz constant Kfy,z such that f. (0, 0, 0) is bounded. Assume that one of the following conditions holds: 1. (finite activity) f has property (Af in ). e is a BMO(P )-martingale. 2. (infinite activity) f has property (Ainf i ) and U ∗ µ

Then |Yt | ≤ exp(Kfy,z (T − t))(|ξ|∞ + (T − t)|f· (0, 0, 0)|∞ ) holds for all t ≤ T , and in particular |Y |∞ ≤ exp(Kfy,z T )(|ξ|∞ + T |f· (0, 0, 0)|∞ ). Proof. This follows directly from Proposition 1.12 and Example 1.10, since f satisfies condition (Aγ ) (resp. (A0γ )) using equation (1.13).

Section 1.2. Comparison theorems and a-priori-estimates

Page 31

In the last part of this section we apply our comparison theorem for more concrete generators. To this end, we consider a truncation fe of a generator f at truncation bounds a < b (depending on time only), given by fet (y, z, u) := ft (κ(t, y), z, κ(t, y + u) − κ(t, y)),

(1.14)

with κ(t, y) := (a(t) ∨ y) ∧ b(t). Next, we show that if a generator satisfies (Aγ ) within the truncation bounds, then the truncated generator satisfies (Aγ ) everywhere. Lemma 1.14. Let f satisfy (1.12) for Y, U such that a(t) ≤ Yt− , Yt− + Ut (e), Yt− + Ut0 (e) ≤ b(t),

t ∈ [0, T ],

e). and let γ satisfy one of the conditions of Example 1.6 for the martingale property of E(γ ∗ µ e Then f satisfies (1.12). Especially, if f satisfies (Af in ) on the set where a(t) ≤ y, y + u ≤ b(t) then fe is Lipschitz in y and z, locally Lipschitz in u and satisfies (Aγ ).

Proof. Indeed, we have fet (Yt− , Zt , Ut ) − fet (Yt− , Zt , Ut0 ) = ft (κ(t, Yt− ), Zt , κ(t, Yt− + Ut ) − κ(t, Yt− )) − ft (κ(t, Yt− ), Zt , κ(t, Yt− + Ut0 ) − κ(t, Yt− )) ≤ ≤

Z ZE E

γ t (e)(κ(t, Yt− + Ut (e)) − κ(t, Yt− + Ut0 (e))) ζt (e) λ(de) γ t (e)(1{γ≥0,U ≥U 0 } + 1{γ −1 , (t, ω) ∈ [0, T ] × Ω.

(2.10)

Section 2.2. Good-deal valuation and hedging

Page 62

For a correspondence C satisfying (2.10) we will associate a closed-convex-valued correspondence Ce with values n

Cet (ω) := (γ, β) ∈ L2 (λ) × Rn :

o  γ 1{ζt (ω)>0} , β ∈ C¯t (ω) , ζt (ω)

p

(2.11)

in the Hilbert space L2 (λ) × Rn , where C¯t (ω) is the closure of Ct (ω) in L2 (λt (ω)) × Rn , (t, ω) ∈ [0, T ] × Ω. Let P denote the completion of the predictable σ-field P under the measure P ⊗ dt. For the application of standard measurable selection arguments in our general setting, we will need the following Assumption 2.9. The correspondence Ce associated to C by (2.11) is P-measurable. Let us note at first that Assumption 2.9 will be satisfied (e.g. in Lemma 2.22 and Lemma 2.30) for some concrete examples of no-good-deal constraint correspondences and general measures λ. Also note that assuming P-measurability is weaker than assuming predictability. The definition of measurability of a correspondence is in the sense of [AF90]: ofor each closed n 2 n −1 e set F ⊂ L (λ) × R , the set C (F ) := (t, ω) ∈ [0, T ] × Ω : Cet (ω) ∩ F 6= ∅ is measurable. Note that completeness of the underlying σ-field is usually required in the theory of measurable correspondences with values in an infinite-dimensional spaces. Since our correspondence Ce by definition assumes values in L2 (λ) × R for a possibly infinitely-supported measure λ on R, it appears natural to require Assumption 2.9 for P instead of P. For correspondences taking values in a finite dimensional space, completeness of the underlying σ-field is not necessary. For the theory of measurable correspondences and existence of measurable selection, we refer to [AF90, Chapter 8] in infinite dimensional spaces, and to [Roc76] in finite dimensional space. For measures λ that are finitely-supported (e.g. for finite state semi-Markov processes) we will therefore rely on results of [Roc76]. The particularity of Ce in comparison to C is that the range of Ce does not depend on (t, ω), and it will be useful for applying measurable selection arguments (e.g. in the proof of Lemma 2.14). e e) is a positive For predictable β and P-measurable γ such that Γγ,β := E (β · W + γ ∗ µ γ,β the probability measure equivalent to uniformly integrable martingale, we denote by Q P with Girsanov kernels (γ, β), i.e. with density process Γγ,β . For risk-neutral measures Qγ,β ∈ Me , the martingale condition of Lemma 2.8 additionally requires that β = −ξ + η with ηt (ω) ∈ Ker σt (ω), (t, ω) ∈ [0, T ] × Ω. Hence we define the set Qngd := Qngd (S) of no-good-deal pricing measures as n

o

Qngd := Qγ,β ∼ P : (γ, β) ∈ C, β = −ξ + η, η ∈ Ker σ ⊆ Me ,

(2.12)

where we do (implicitly) require that Γγ,β is a uniformly integrable martingale to define probability measures Qγ,β . Assume (0, −ξ) ∈ C,

(2.13)

Section 2.2. Good-deal valuation and hedging

Page 63

b γ ,β b = (0, −ξ). In (2.12) we implicitly b = Qb which implies in particular Q ∈ Qngd 6= ∅ since (γb , β) required the γ-components of Girsanov kernels to satisfy γt ∈ L2 (λt ) for any t ∈ [0, T ]; see (2.10). Note that such a restriction is also made indirectly by [BS06], where the constraint on the size of the instantaneous Sharpe ratios via the Hansen-Jagannathan bounds implicitly requires the L2 (λt )-norm of γt to be finite, for any t ∈ [0, T ]. For instance the latter holds e is a locally square integrable local martingale such that |γ|2 ∗ ν is locally integrable if γ ∗ µ (cf. [JS03, Chapter II, Section 1]). Under uniform boundedness of the correspondence C (see e will be automatically satisfied. Section 2.3), the local square integrability of γ ∗ µ

For sufficiently integrable contingent claims X, the upper and lower good-deal valuation bounds are defined by πtu (X) := ess sup EtQ [X] Q∈Qngd

and

πtl (X) := ess inf EtQ [X], Q∈Qngd

t ∈ [0, T ].

(2.14)

Because π l (X) = −π u (−X), we focus on studying the upper good-deal bound. Recall the property of multiplicatively stable (shortly m-stable) sets Q of probability measures Q ∼ P : for all Γ1 , Γ2 ∈ Q and τ ≤ T stopping time, the process Γ with Γt := 1{t≤τ } Γ1t + 1{τ ≤t} Γ1τ Γ2t /Γ2τ belongs to Q, or equivalently the random variable ΓT := Γ1τ Γ2T /Γ2τ defines an element of Q. The following result of [Del06] (stated here as in [KS07b, Theorem 2.7] or [Bec09, Proposition 2.6]) provides good dynamic properties for suprema of conditional expectations over an m-stable set of equivalent measures. Lemma 2.10. Let Q be a convex and m-stable set of measures Q ∼ P and πtu,Q (X) := ess supQ∈Q EtQ [X], t ∈ [0, T ], X ∈ L∞ . Then for all X ∈ L∞ there exists a c`ad`ag version Y of π·u,Q (X) such that Yτ = ess supQ∈Q EτQ [X] =: πτu,Q (X) for any stopping time τ ≤ T . Moreover π·u,Q (·) satisfies the properties of a dynamic coherent risk measure. It is recursive and time consistent: For all σ ≤ τ ≤ T stopping times and for all X1 , X2 ∈ L∞ , πσu,Q (πτu,Q (X 1 )) = πσu,Q (X 1 ) and πτu,Q (X 1 ) ≥ πτu,Q (X 2 ) implies πσu,Q (X 1 ) ≥ πσu,Q (X 2 ). Finally, a supermartingale property holds: for all Q ∈ Q and for all stopping times σ ≤ τ ≤ T ,   πσu,Q (X) ≥ EσQ πτu,Q (X) . In particular, π·u,Q (X) is a supermartingale under any Q ∈ Q. One can show that the sets Qngd , Me are convex and m-stable, enabling an application of ngd Lemma 2.10 to the good-deal bound π·u (X) = π·u,Q (X). That Me := Me (S) is m-stable and convex is a consequence of [Del06, Proposition 5]. The next result shows that Qngd as defined in (2.12) is also convex and m-stable; the proof is included in the appendix. Lemma 2.11. The set Qngd is convex and m-stable. We use a notion of good-deal hedging similar to the one in [Bec09] (see also Chapter 3), where a hedging strategy arises as the minimizer of a suitable a-priori dynamic risk measure of no-good-deal type, for which the minimal capital to make the position acceptable coincides with

Section 2.2. Good-deal valuation and hedging

Page 64

the good-deal valuation bound. The dynamic risk measure to be minimized over all permitted trading strategies is defined for sufficiently integrable contingent claims X, as ρt (X) = ess sup EtQ [X],

t ∈ [0, T ],

Q∈P ngd

where P ngd is a set of a-priori valuation measures to be chosen so that, in the spirit of [BE09], the good-deal bound π·u (X) becomes the market consistent risk measure (valuation) for X that arises from no-good-deal hedging with respect to ρ. An investor holding a liability X and trading parallely in the market according to a permitted strategy φ ∈ Φ would assign to R c her position at every time t ∈ [0, T ] the risk ρt (X − tT φtr s dWs ). For optimal trading, she would like to use a strategy φ¯ ∈ Φ that minimizes her risk at any time t ∈ [0, T ], in such a way that the minimal capital requirement to make her position ρ-acceptable is πtu (X). This minimum yields the market-based risk measure (after optimal risk-sharing with the financial market) in the spirit of [BE09]. In other words, good-deal valuation should arise from good-deal hedging by minimizing the dynamic coherent risk measure ρ with respect to the family P ngd of a-priori measure as generalized scenarios (see [ADE+ 07]). This yields a hedging notion that corresponds to good-deal valuation, in that the market -based risk measure turns out to be the good-deal valuation bound π·u (X). The investor’s hedging problem therefore is to find φ¯ ∈ Φ such that 

πtu (X) = ρt X −

Z t

T





c φ¯tr s dWs = ess inf ρt X − φ∈Φ

Z t

T



c φtr s dWs , t ∈ [0, T ].

(2.15)

Since 0 ∈ Φ, then (2.15) necessarily requires πtu (X) ≤ ρt (X), t ∈ [0, T ], which in turns hints that P ngd should contain the smaller set Qngd . As in [Bec09] we choose P ngd as the set of probability measures equivalent to P , that are not necessarily martingale measures and satisfy the no-good-deal constraint with C, i.e. such that Qngd = P ngd ∩ Me . More precisely we define n o P ngd := Qγ,β ∼ P : (γ, β) ∈ C . (2.16) In a financial market with no risky asset, i.e. S ≡ 1, any probability measure is a martingale measure and consequently P ngd defined in (2.16) coincides with Qngd (1). Hence P ngd inherits the m-stability and convexity of Qngd (1), and thus ρ· (·) is satisfies the properties in Lemma 2.10: ρ is a dynamic coherent time-consistent risk measure. For a contingent claim X, the tracking error Rtφ (X) from hedging according to a strategy φ ∈ Φ, at time t ∈ [0, T ], is defined by the difference between the capital variations of the claim and the profit/loss from dynamic trading up to time t according to φ, i.e. ct . Rtφ (X) := πtu (X) − π0u (X) − φ · W c , with x0 ∈ R, Remark 2.12. For a self-financing strategy φ ∈ Φ replicating X = x0 + 0T φtr dW φ the tracking error vanishes, i.e. R (X) = 0. One says that a strategy is a mean-self-financing R

Section 2.3. Case of uniformly bounded correspondences

Page 65

(like risk-minimizing strategies studied in [Sch01, Section 2], with EtQ [X] taking the role of πtu (X)) if its tracking error it is a martingale (under P ). We will show (see Theorem 2.19) that the tracking error of a good-deal hedging strategy φ¯ satisfying (2.15) is a supermartingale under all a-priori measures in P ngd . The result will therefore enable us to view φ¯ as being “at least mean-self-financing” under any Q ∈ P ngd . This will be seen as a robustness property of φ¯ with respect to the set of measures P ngd interpreted as generalized scenarios (in the sense of [ADE+ 07]). b

For results on good-deal valuation and hedging, we shall distinguish two cases, namely the case where the constraint correspondence is uniformly bounded, and the case beyond uniform boundedness. For the first case we will obtain descriptions of good-deal bounds and hedging strategies in terms of solutions to Lipschitz JBSDEs. In the second case, Lipschitz JBSDE tools are not directly applicable, and we will resort to approximation arguments focusing only on valuation.

2.3

Case of uniformly bounded correspondences

We characterize π u (X) as solution to a JBSDE under the assumption that C is uniformly bounded, which ensures that the resulting JBSDE has a Lipschitz generator function. Indeed, the connection between dynamic coherent risk measures and BSDEs is quite known; cf. e.g. [Ros06, PR15]. We say that a correspondence C satisfying (2.10) is uniformly bounded if Assumption 2.13. sup(t,ω) sup(γ,β)∈Ct (ω) (kγk2L2 (λt (ω)) + |β|2 ) < +∞. Under Assumption 2.13 one can show as in [QS13, Proposition 3.2] that Γ ∈ S 2 for any Γ density processes of a measure in P ngd . Hence for contingent claims X ∈ L2 ⊃ L∞ that may be path-dependent, πtu (X) and ρt (X) are well-defined as essential suprema of almost surely finite-valued random variables, and one can check (also for X ∈ L2 ) that an analogue of Lemma 2.10 still holds. For each no-good-deal measure Q ∈ Qngd , Lemma 2.4 describes E·Q [X] as the value process of a JBSDE with linear generator. Then using the comparison principle Proposition 2.6, one can describe π·u (X) (and likewise for ρ· (X)) as the value process of a JBSDE whose generator is the supremum of the linear ones. The following Lemma 2.14 (see Appendix 2.5 for its proof using Assumption 2.13) that the maximum is indeed attained. This yields (cf. Theorem 2.16) a worst-case measure under which the good-deal bound is attained. Obviously such a worst-case measure will usually lie in the L1 -closure of the set Qngd of no-good-deal measures. Lemma 2.14. Let Assumptions 2.9 and 2.13 hold for C satisfying (2.10). Let C¯t (ω) denote the closure of Ct (ω) in L2 (λt (ω)) × Rn , (t, ω) ∈ [0, T ] × Ω and let Z ∈ H2 and U ∈ Hν2 . Then

Section 2.3. Case of uniformly bounded correspondences

Page 66

e a) there exist η¯ = η¯(Z, U ) predictable and γ¯ = γ¯ (Z, U ) P-measurable such that for P ⊗ dtalmost all (ω, t) ∈ Ω × [0, T ] holds

(¯ γt (ω), η¯t (ω)) ∈ argmax ηttr (ω)Π⊥ (t,ω) (Zt (ω)) + (γt (ω),ηt (ω))

Z E

Ut (ω, e)γt (ω, e)λt (ω, de), (2.17)

with the supremum taken over all (γt (ω), ηt (ω)) with (γt (ω), −ξt (ω) + ηt (ω)) ∈ C¯t (ω) and ηt (ω) ∈ Ker σt (ω). e ˜ b) there exist β˜ = β(Z, U ) predictable and γ˜ = γ˜ (Z, U ) P-measurable such that for P ⊗ dtalmost all (ω, t) ∈ Ω × [0, T ] holds

(˜ γt (ω), β˜t (ω)) ∈

argmax ¯t (ω) (γt (ω),βt (ω))∈C

βttr (ω)Zt (ω)

Z

Ut (ω, e)γt (ω, e)λt (ω, de).

+ E

To (¯ γ , β¯ := −ξ + η¯) ∈ C¯ of Part a) of Lemma 2.14, we associate the probability measure ¯ := Qγ¯,β¯  P , which might not be equivalent to P since γ¯ may take the value −1 on a Q ¯ is possibly not an equivalent local martingale measure, but we now non-negligible set. So Q show that it belongs to the L1 (P )-closure of Qngd . Lemma 2.15. For Z ∈ H2 and U ∈ Hν2 , let (¯ γ , η¯) be as in Part a) of Lemma 2.14. Define ¯ γ ¯ , β b + (1 − 1 )Q, ¯ ¯ ¯ for all n ∈ N. the measures Q = Q  P for β := −ξ + η¯ and Qn := n1 Q n n ngd n 1 ¯ in L (P ) as n → ∞. Consequently, it holds Then (Q )n∈N ⊂ Q and Q converges to Q ¯ Q u ¯ πt (X) ≥ E [X], Q-a.s., t ∈ [0, T ]. t

Proof. Let n ∈ N. Clearly Qn ∼ P . Moreover dQn /dP = Z n := n1 Zb + (1 − n1 )Z¯ with Zb := b ¯ dQ/dP = E(−ξ · W ) and Z¯ := dQ/dP . Itˆo formula then yields Z n = E((−ξ + η n ) · W + γ n ∗ µ ˜) n n e ¯ n ∈ [0, 1) for η = α¯ η being predictable and γ = α¯ γ is P-measurable with α = (1 − n1 )Z/Z thanks to Zb > 0. Therefore η n ∈ Ker σ and γ n > −1 due to γ¯ ≥ −1. Hence (γ n , η n ) ∈ C and n n ¯ in L1 (P ) as n → ∞ is straightforward so Qn = Qγ ,η is in Qngd . Convergence of Qn to Q ¯ by definition of Qn and this implies πtu (X) ≥ EtQ [X] for all t ≤ T . For X ∈ L2 , consider the two JBSDEs: Z

T

Yt = X +



(−ξs + η¯s )tr Zs +

t



Z t

Z



Us (e)¯ γs (e)λs (de) ds (2.18)

E T

Zstr dWs



Z

T

Z

t

E

e(ds, de), Us (e)µ

and Z

T

Yt = X + t



Z t

T



Zstr β˜s +

Zstr dWs

Z





Us (e)˜ γs (e)λs (de) ds (2.19)

E

Z t

T

Z E

e(ds, de), Us (e)µ

Section 2.3. Case of uniformly bounded correspondences

Page 67

˜ = (˜ ˜ with (¯ γ , η¯) = (¯ γ (Z, U ), η¯(Z, U )) and (˜ γ , β) γ (Z, U ), β(Z, U )) being by Lemma 2.14. Then we have the following Theorem 2.16. Let Assumptions 2.9 and 2.13 hold for C satisfying (2.10). Let X ∈ L2 , and ˜ be as in Lemma 2.14. Then (¯ γ , η¯) and (˜ γ , β) ¯P a) the JBSDE (2.18) has a unique solution (Y, Z, U ) in S 2 × H2 × Hν2 and there exists Q  ¯ with density dQ/dP = E (−ξ + η¯) · W + γ¯ ∗ µ ˜ such that π·u (X) satisfies ¯

πtu (X) = ess sup EtQ [X] = Yt = EtQ [X],

¯ Q-a.s., t ∈ [0, T ].

Q∈Qngd

˜ U ˜ ) in S 2 × H2 × H2 and there exists b) the JBSDE (2.19) has a unique solution (Y˜ , Z, ν  e  P with dQ/dP e Q = E β˜ · W + γ˜ ∗ µ ˜ such that e ρt (X) = ess sup EtQ [X] = EtQ [X] = Yt , Q-a.s., t ∈ [0, T ]. e

Q∈P ngd

For X ∈ L2 and permitted trading strategies φ ∈ Φ, consider the JBSDE T

Z

Yt = X + t



T

Z t



− ξstr φs + (Zs − φs )tr β˜s (Z − φ, U ) +

Zstr dWs −

Z t

T

Z



Us (e)˜ γs (Z − φ, U )(e)λt (de) ds

E

Z E

(2.20)

e(ds, de), Us (e)µ

where for φ ∈ Φ the processes γ˜· (Z − φ, U ) and β˜· (Z − φ, U ) are as in Part b) of Lemma 2.14. Then we have the following lemma, whose proof is deferred to Appendix 2.5. Lemma 2.17. Let Assumptions 2.9 and 2.13 hold for C satisfying (2.10). For X ∈ L2 and φ ∈ Φ, the (2.20) admits a unique solution (Y φ , Z φ , U φ ) ∈ S 2 × H2 × Hν2 that satisfies  JBSDE R c Ytφ = ρt X − tT φtr s dWs , t ∈ [0, T ]. Let f and f φ ( for φ ∈ Φ) denote respectively the generators of the JBSDEs (2.18) and (2.20), given for z ∈ Rn , u ∈ L2 (λt ), t ∈ [0, T ] as f (t, z, u) =

ess sup

tr

Z

β z+

u(e)γ(e)λt (de)

¯t (γ,β)∈C β∈−ξt +Ker σt φ

f (t, z, u) =

−ξttr φt

E

+ ess sup β (z − φt ) + tr

¯t (γ,β)∈C

Z



and 

u(e)γ(e)λt (de) . E

The following lemma will be used in combination with the comparison theorem for JBSDEs to show existence of a good-deal hedging strategy φ¯ solution to the hedging problem (2.15). The proof is also deferred to the Appendix 2.5.

Section 2.3. Case of uniformly bounded correspondences

Page 68

Lemma 2.18. Let Assumptions 2.9 and 2.13 hold for C satisfying (2.10). Then for all z ∈ Rn , u ∈ L2 (λt ), t ∈ [0, T ] holds (2.21)

f (t, z, u) = ess inf f φ (t, z, u). φ∈Φ

To prove a general existence result for φ¯ solution to the hedging problem (2.15), we will require the additional condition on the abstract correspondence C that there exists  ∈ (0, 1) such that {0} × B (−ξt (ω)) ⊆ Ct (ω), for all (t, ω),

(2.22)

where B (−ξ) denotes the (closed or open) ball in Rn centered at −ξ with radius . Condition (2.22) implies in particular (2.13), and will be automatically satisfied for concrete no-good-deal constraint correspondences as in the frameworks of Section 2.3.1 and Section 2.4.2. In addition, (2.22) ensures coercivity of the generators of the JBSDEs (2.20) as functions of φ ∈ Φ, i.e. f φ (t, z, u) → +∞ as |φ| → ∞ for fixed (t, z, u). Following common arguments in variational analysis, (2.22) will be used in the proof of Theorem 2.19 below to deduce existence of a minimizing strategy φ¯ in (2.15). The precise result is the following, and shows moreover that hedging strategies φ¯ are at least mean-self-financing in the sense that their tracking errors satisfy a supermartingale property with respect to measures in P ngd . The proof is postponed to Appendix 2.5. Theorem 2.19. Let Assumptions 2.9 and 2.13 hold for C satisfying (2.10) and (2.22). For X ∈ L2 , let (Y, Z, U ), (Y φ , Z φ , U φ ) in S 2 × H2 × Hν2 (for φ ∈ Φ) be solutions to the γ , η¯), (˜ γ , η˜) as in Lemma 2.14. Then there exists JBSDEs (2.18), (2.20) respectively, with (¯ ¯ ¯ φ := φ(X) ∈ Φ satisfying ¯

f φ (t, Zt , Ut ) = ess inf f φ (t, Zt , Ut ), t ∈ [0, T ], φ∈Φ

¯ and for such φ¯ hold Yt = ess inf Ytφ = Ytφ , t ∈ [0, T ], and φ∈Φ

πtu (X)



= ess inf ρt X − φ∈Φ

Z t

T

c φtr s dWs





= ρt X −

Z t

T



c φ¯tr s dWs = Yt ,

t ∈ [0, T ].

(2.23)

¯ Moreover the tracking error Rφ (X) of φ¯ is a Q-supermartingale for any Q ∈ P ngd and a ∗ ∗ ¯ U ), β(Z ˜ − φ, ¯ U )). Q∗ -martingale, for Q∗ = Qγ ,β ∈ P ngd with (γ ∗ , β ∗ ) := (˜ γ (Z − φ,

Remark 2.20. 1. In accordance with Remark 2.12, Theorem 2.19 shows that good-deal hedging strategies are robust in the sense that they are at least mean-self-financing with respect to the set P ngd as generalized scenarios. 2. Note that Theorem 2.19 shows only existence of φ¯ and does not claim its uniqueness. The latter may depend on the contingent claim X into consideration. Independently

Section 2.3. Case of uniformly bounded correspondences

Page 69

of the contingent claim, uniqueness may be obtained for particular structures of the correspondence C. We will provide examples (see Section 2.3.2 and last example of Section 2.3.1), where uniqueness is ensured for any claim, and explicit expressions of φ¯ in terms of JBSDE solutions can be obtained. If the values of the correspondence C can be decomposed as Ct = Ctγ × Ctβ for Ctγ ⊂ L2 (λt ) and Ctβ ⊂ Rn , t ∈ [0, T ], i.e. the no-good-deal constraint decouples as a constraint on the unpredictable event-risk separated from a constraint on the market price of stock risk, then the hedging strategy φ¯ does not depend on the U -component of the solution to the JBSDE (2.18). The precise statement is summarized in the following corollary of Theorem 2.19. Corollary 2.21. Let the conditions of Theorem 2.19 hold and assume in addition that Ct = Ctγ × Ctβ for Ctγ ⊂ L2 (λt ) and Ctβ ⊂ Rn , t ∈ [0, T ]. For X ∈ L2 , let (Y, Z, U ) be the solution to the JBSDE (2.18). Then a good-deal hedging strategy is given by φ¯ ∈ Φ satisfying φ¯t ∈ argmin φ∈Φ

2.3.1





− ξttr φt + ess sup βttr (Zt − φt ) , t ∈ [0, T ]. ¯β βt ∈ C t

Results for constraint on instantaneous Sharpe ratios (bounded case)

We consider a no-good-deal constraint emanating from a bound on the instantaneous Sharpe ratios of investments in the financial market extended by additional derivative price processes. Recall that this case was studied in [BS06] for a Markovian model of asset prices and additional factor processes exhibiting jumps. [BS06, Theorem 2.3] showed an extended form of the Hansen–Jagannathan (HJ) inequality in the sense that |SRt | ≤ k(γt , βt )kL2 (λt )×Rn , t ∈ [0, T ], for all (γ, β) Girsanov kernels of measures in Me , where SRt denotes the instantaneous Sharpe ratio at time t. This meant that a bound on the instantaneous Sharpe ratios could be achieved through a bound on the norm of the Girsanov kernels of pricing measures and an application of the HJ inequality. Their no-good-deal constraint was then set as k(γt , βt )k2L2 (λt )×Rn := kγt k2L2 (λt ) + |βt |2 ≤ K 2 ,

(2.24)

t ∈ [0, T ],

for some given K ∈ (0, ∞), and they derived the good-deal bounds in terms of solutions to HJB PIDEs using dynamic programming techniques. Here we rather use JBSDEs to derive good-deal bounds in a non-necessarily Markovian model under a no-good-deal constraint of the type (2.24), but for more general K being a positive bounded predictable process. The associated constraint correspondence C is then n

o

Ct = (γ, β) ∈ L2 (λt ) × Rn : γ > −1, kγk2L2 (λt ) + |β|2 ≤ Kt2 .

(2.25)

Beyond the boundedness of ξ, we assume that for some ε > 0 holds Kt > |ξt | + ε,

t ∈ [0, T ],

(2.26)

Section 2.3. Case of uniformly bounded correspondences

Page 70

b ∈ Qngd 6= ∅. Since K is bounded and predictable, the so that (0, −ξ) ∈ C and hence Q no-good-deal restriction in this example fits well into the framework of the current Section 2.3 since C given by (2.25) is then convex-valued, and satisfies (2.10) and Assumption 2.13. Were K not uniformly bounded, then C might fail to satisfy Assumption 2.13, and the framework of the upcoming Section 2.4 would then prevail. Here the closed-valued correspondence C¯ is given by n o C¯t = (γ, β) ∈ L2 (λt ) × Rn : γ ≥ −1, kγk2L2 (λt ) + |β|2 ≤ Kt2 . (2.27)

Indeed for arbitrary (γ, β) ∈ C¯t (ω), one chooses the sequence (γ k , β k )k∈N ⊂ Ct (ω) with γ k = γ ∨ (−1 + k1 ) and β k = β, so that γ k ≤ |γ| holds (since γ ≥ −1). By dominated convergence, it then follows that (γ k , β k ) converges to (γ, β) in L2 (λt (ω)) × Rn . The correspondence Ce defined as in (2.11) is then given by n

1/2

Cet = (γ, β) ∈ L2 (λ) × Rn : γ ≥ −ζt

o

, kγk2L2 (λ) + |β|2 ≤ Kt2 .

(2.28)

Applying the theory of measurable correspondences in [AF90], the following lemma shows that e The proof is relegated to Appendix 2.5. Assumption 2.9 holds for C. Lemma 2.22. The closed-convex-valued correspondence Ce given by (2.28) is P-measurable. Now one can apply part a) of Theorem 2.16 to obtain a description of π·u (X) and a worst-case no-good-deal measure for X ∈ L2 in terms of the solution to the JBSDE (2.18). The precise result is the following Theorem 2.23. For X ∈ L2 , the JBSDE (2.18) with (¯ γ , η¯) from (2.17) has a unique solution 2 2 2 ¯ (Y, Z, U ) in S × H × Hν . Moreover there exists Q  P in the L1 -closure of Qngd (in ¯ the sense of Lemma 2.15), with density dQ/dP = E ((−ξ + η¯) · W + γ¯ ∗ µ ˜) such that the u good-deal bound π· (X) satisfies ¯

πtu (X) = ess sup EtQ [X] = Yt = EtQ [X]

for all t ≤ T.

Q∈Qngd

(2.29)

Sufficient conditions for explicit formulas for valuation and hedging Let us still consider the no-good-deal constraint on the Sharpe ratio described in terms of the correspondence C in (2.25). We investigate conditions ensuring an explicit form of the maximizer (¯ γ (Z, U ), η¯(Z, U )) in the generator of the JBSDE (2.18), which in turn may also lead to an explicit formula for the good-deal hedging strategy. Note that the classical Kuhn-Tucker routine may not apply for the maximization problem in (2.17) for C in (2.25), due to the additional constraint {γ ≥ −1}. If one considers the good-deal valuation problem without this constraint for JBSDEs, then can obtain using Kuhn-Tucker arguments an explicit maximizer

Section 2.3. Case of uniformly bounded correspondences

Page 71

for all t ∈ [0, T ] as 2 1/2 Kt2 − |ξt | ⊥ and η¯t = 2 1/2 Πt (Zt ). ⊥ 2 kUt kL2 (λt ) + Πt (Zt )

2 1/2 Kt2 − |ξt | γ¯t = 2 1/2 Ut kUt k2L2 (λt ) + Π⊥ t (Zt )

(2.30)

In general the relaxed Girsanov kernels in (2.30) do not induce a measure Qγ¯,−ξ+¯η that is absolutely continuous with respect to P . In addition, (¯ γ , η¯) from (2.30) only give rise to a u,r u relaxed bound π (X) which is clearly larger than π (X), i.e. π u,r (X) ≥ π u (X), for any financial risk X since it is obtained by maximizing EtQ [X] over a set of measure Qr ⊇ Qngd containing eventually signed measures. These facts were already analyzed in [BS06, Section 3.5 and 4.4], where similar relaxed good-deal bounds was studied using Hamilton-Jacobi-Bellman techniques. In terms of JBSDEs, we obtain here the following relaxed version of the JBSDE (2.18), with an explicit generator and value process π·u,r (X) (instead of π·u (X)) by replacing (¯ γ , η¯) in (2.18) by the expressions in (2.30)): Z

T

Yt = X + t



Z t

T



− ξstr Πs (Zs ) + Ks2 − |ξs |2

Zstr dWs −

T

Z t

1/2

2 kUs k2L2 (λs ) + |Π⊥ s (Zs )|

Z E

1/2 

ds (2.31)

e(ds, de). Us (e)µ

The JBSDE (2.31) has a Lipschitz generator, and hence (by e.g. [Bec06, Proposition 3.2]) admits a unique solution (Y, Z, U ) ∈ S 2 × H2 × Hν2 , for X ∈ L2 , with π·u,r (X) := Y . Note that for a Sharpe ratio bound K conveniently chosen (for example small enough), the relaxed good-deal bound π·u,r (X) could still be lower than the upper no-arbitrage bound. However in general π·u,r (X) may not be a no-arbitrage price since (¯ γ , −ξ + η¯) in (2.30) could define a signed measure because γ¯ ≥ −1 may be violated. If for a contingent claim X ∈ L2 one can show that U ≥ 0 for (Y, Z, U ) solving the JBSDE (2.31), then γ¯ in (2.30) satisfies γ¯ ≥ 0 > −1 and the generators of the JBSDEs (2.18) and (2.31) coincide. This would clearly imply that π·u (X) = π·u,r (X), and both are described by the solution of the JBSDE (2.31). In this case, it would even possible to obtain a closed-form ¯ Precisely, we have the following expression of good-deal hedging strategies φ. Proposition 2.24. Assume the Sharpe ratio constraint described by the correspondence C in (2.25). For X ∈ L2 , let (Y, Z, U ) be the unique solution to the JBSDE (2.31). Then 1. If the good-deal bound and its relaxed version coincide, i.e. π·u (X) = π·u,r (X), then a good-deal hedging strategy φ¯ ∈ Φ is given by 

φ¯t =

2 1/2



kUt k2L2 (λt ) + Π⊥ t (Zt ) Kt2 − |ξt |2

1/2

ξt + Πt (Zt ), t ∈ [0, T ].

(2.32)

Section 2.3. Case of uniformly bounded correspondences

Page 72

¯ ¯ = Qγ¯,−ξ+¯η 2. If U ≥ 0, then πtu (X) = πtu,r (X) = EtQ [X] = Yt , t ∈ [0, T ], where Q

is in Qngd with (¯ γt , η¯t ) = Kt2 − |ξt |2 satisfying γ¯ ≥ 0 > −1.

1/2



2   −1/2

kUt k2L2 (λt ) + Π⊥ t (Zt )



U t , Π⊥ t (Zt ) ,

Proof. The second claim follows from the preceding discussion. As for the first claim, if π·u (X) = π·u,r (X) then we have for any t ∈ [0, T ] ¯ f φ (t, Zt , Ut ) = −ξttr φ¯t + ess sup β tr (Zt − φ¯t ) + ¯t (γ,β)∈C





Z

γ(e)Ut (e)λt (de) E

2 1/2

≤ −ξttr φ¯t + Kt kUt k2L2 (λt ) + Zt − φ¯t = −ξttr Πt (Zt ) + Kt2 − |ξt |2

1/2 

2 kUt k2L2 (λt ) + |Π⊥ t (Zt )|

1/2

= f (t, Zt , Ut ). ¯

By Lemma 2.18 it follows that f φ (t, Zt , Ut ) = ess infφ∈Φ f φ (t, Zt , Ut ), t ∈ [0, T ], and therefore applying Theorem 2.19 proves the required claim. A condition similar to U ≥ 0 for obtaining an explicit generator of the JBSDE describing the good-deal bound was provided in [Del12]. However [Del12] focused on a financial/insurance market with a single traded risky asset modelled by a two dimensional Brownian motion (i.e. d = 1, n = 2), and with the random measure µ associated to jumps of a point-process with state-independent compensator of the form ν(dt) = ζt dt for a predictable process ζ ≥ 0. Our result deals with fairly general jump processes and financial markets (with d ≤ n). A drawback of the condition U ≥ 0 in part 2. of Proposition 2.24 which ensures equality between π·u (X) and π·u,r (X) is that it might not be straightforward to check in general, and also depends on the contingent claim X into consideration. Examples of such claims includes derivatives X that solely on diffusive risk and for which one would expect that U = 0. Also, if µ is the random measure of the jumps of a simple Poisson process and X a claim that pays nothing if the Poisson process does not jump before maturity and a unit if it does, then one U would be non-negative. Recall that for π·u (X) and π·u,r (X) to coincide, it is sufficient to provide conditions that guarantee that the process γ¯ defined in (2.30) satisfies γ¯ ≥ −1. If the support of the measure λ is finite, one could write a condition on K and λ that does not depend on the claim and, without further hypotheses on U , ensures that γ¯ ≥ −1 holds. Note that in this case, the controls in the optimization problem in Lemma 2.14 would live in a finite dimensional space, simplifying considerably the problem. To explain what happens in this case, we provide an example in a semi-Markov jump setup that includes continuous time Markov chains.

Section 2.3. Case of uniformly bounded correspondences

Page 73

Example for semi-Markov jump-dynamics: We consider a Markov renewal process as (Jn , Tn )n∈N0 , with random variables (Jn )n taking values in the finite state space E = {e1 , . . . , em } ⊂ Rm (w.l.o.g. to fit into our setup from Section 2.1), for ei for i ∈ I := {1, . . . , m} denoting the canonical unit vectors, with m ∈ N, and a non-decreasing sequence (Tn )n , T0 = 0, of F-measurable [0, T ]-valued random times modeling its renewal times. We assume that the process (Jn , Tn )n starts almost-surely at a pre-specified state and has a stochastic semi-Markov kernel Q = (Qij )i,j∈I exogenously given such that Qij t = P [Jn+1 = ej , Tn+1 − Tn ≤ t | Jn = ei ], t ∈ [0, T ], for all n ∈ N. The process J deP fined by Jt := JNt , where Nt := ∞ n=1 1{Tn ≤t} is the counting process associated to the jumps of (Jn , Tn )n , is called the semi-Markov process associated to the Markov-renewal process. Note that (Jn )n is a Markov chain with transition Kernel P = (pij )i,j∈I satisfying ij ij ij pij = Qij T . It is well-known that the conditional independence relation Qt = p Gt holds, where Gij t = P [Tn+1 − Tn ≤ t | Jn = ei , Jn+1 = ej ] , for all n ∈ N, is the conditional distribution function of the sojourn time Tn+1 − Tn from state ei to state ej . For unexplained notions in the theory of Markov renewal processes, semi-Markov processes and point processes in general, we refer to [C ¸ 75, Bre81]. Let µ be the random measure of the jumps of the semi-Markov process J, identified with P the family of counting processes (N ij )i,j∈I, j6=i through µ([0, t) × D) = i,j∈I, ej −ei ∈D Nti,j , where Ntij =

P∞

n=1 1 Tn ≤t, Jn =ei , Jn+1 =ej



is the number of jumps up to time t from state

ei to state ej of J. Note that Nt = i,j∈I, j6=i Ntij . Let us denote by τt := sup{s ∈ [0, t] : Jt−u = Jt , u ∈ [t − s, t]} the time the process J has spent in its current state Jt . Then the compensator νij of N ij is given by νij (dt) = 1{Jt− =ei } αtij (τt− )dt, with jump intensities P

αtij (u) = lim

h&0

P [Jt+h = ej | Jt = ei , τt = u] , h

u, t ∈ [0, T ], i, j ∈ I, i 6= j,

(2.33)

e ij of N ij is where we set αii = − j6=i αij , i ∈ I. The compensated jump process N R t ij ij ij ij ij e = N − λ ds, where λ := 1 e with the family N {Jt− =ei } αt (τt− ), t ∈ [0, T ]. We identify µ t t t 0 s ij e )i,j∈I, j6=i and assume that (W, µ e) have the weak predictable representation property with (N respect to (F, P ) in the sense that for every local (F, P )-martingale M , there exists predictable R R P ij 2 processes Z, U = (U ij )i,j∈I, j6=i with 0T |Zs |2 ds < ∞ and i,j∈I, j6=i 0T λij s (Us ) ds < ∞ Rt R P t ij ˜ ij such that Mt = 0 Zs dWs + i,j∈I, j6=i 0 Us dNs , t ∈ [0, T ]. P

For (Z, U ) ∈ H2 × Hν2 , the expressions of γ¯ and η¯ from (2.30) for γ¯ = (¯ γ ij )i∈I,j6=i and ˜ ij become U = (U ij )i∈I,j6=i suitably re-parametrized in terms of integrands with respect to N γ¯tij = Kt2 − |ξt |2

1/2 

2 |Π⊥ t (Zt )| +

X

kl 2 λkl t (Ut )

−1/2

Utij ,

−1/2

Π⊥ t (Zt ).

and

k,l∈I, l6=k

η¯t = Kt2 − |ξt |2

1/2 

2 |Π⊥ t (Zt )| +

X k,l∈I, l6=k

kl 2 λkl t (Ut )

Section 2.3. Case of uniformly bounded correspondences

Page 74

Assume that the process K is small enough such that the inequality K 2 − |ξ|2 1{λij 6=0} ≤ λij , P ⊗ dt-a.s., for all i, j ∈ I, i 6= j,

(2.34)



holds. Since U ∈ Hν2 , we can assume without loss of generality that U ij = 0 on {λij = 0}. With this, (2.34) implies that for any i, j ∈ I, i 6= j, t ∈ [0, T ] hold Kt2 − |ξt |2

1/2

Utij ≥ − Kt2 − |ξt |2

1/2

|Utij | ≥ − λij t 

≥−

1/2

|Utij | ij 2 λkl t |Ut |

X

1/2

k,l∈I, l6=k



2 ≥ − |Π⊥ t (Zt )| +

X

kl 2 λkl t (Ut )

1/2

.

k,l∈I, l6=k

Hence γ¯ ij ≥ −1 for any i, j ∈ I, j 6= i, which in turn ensures π·u (X) = π·u,r (X) = Y for X ∈ L2 , with (Y, Z, U ) solution to the JBSDE (2.31) which now reads Z

T





Yt = X + t



Z

ξstr Πs (Zs )

+

(Ks2

2 1/2

− |ξs | )



2 |Π⊥ s (Zs )|

+

X

ij 2 λij t (Ut )

1  2

ds

i,j∈I, j6=i T

t

Zstr dWs −

X

Z

i,j∈I, j6=i t

T

˜ ij . Utij dN t

(2.35)

Note that (2.34) implies γ¯ ≥ −1 and not γ¯ > −1 in general. While the latter holds when U ≥ 0, already the former is sufficient to ensure that π·u (X) = π·u,r (X). For good-deal hedging, applying part 1. of Proposition 2.24 therefore implies that a good-deal hedging strategy exists for any contingent claim X ∈ L2 and is expressed as

P

φ¯t (X) =

2 1/2

ij ij 2 ⊥ i,j∈I, j6=i λs (Us ) + Πt (Zt )

Kt2 − |ξt |2

1/2

ξt + Πt (Zt ), t ∈ [0, T ].

We now provide conditions under which the BSDE (2.35) can be reduced to a system of ODEs, which may then be solved numerically forward in time. To this end, suppose that the semi-Markov process is a continuous time Markov chain. For A = (αij )i,j∈I denoting the deterministic but time-dependent rate matrix of the chain J with entries P [Jt+h = ej | Jt = ei ] ≥ 0, t ∈ [0, T ], h&0 h

αij (t) = lim

ij for i 6= j, and j∈I αij = 0 for i ∈ I, it follows that λij t := 1{Jt− =ei } α (t), t ∈ [0, T ]. As in [CS12] and analogously to the boundedness assumption on density ζ in the general setup, we assume that the components of the rate matrix process A are uniformly bounded so that with positive probability the Markov chain does not change state on fixed compact time intervals. For this part, we assume that W and J are independent, and F is the augmentation of FW ∨ FJ .

P

Section 2.3. Case of uniformly bounded correspondences

Page 75

Using J as a factor process, assume that the market price of risk ξ and Sharpe ratio bound K are deterministic functions of the chain, i.e. ξt := ξ(t, Jt− ) and Kt := K(t, Jt− ), t ∈ [0, T ], for K : [0, T ] × Rm → (0, ∞) and ξ : [0, T ] × Rm → Rn measurable. For a contingent claim X = G(JT ), depending solely on the final state of the Markov chain, with G : Rm → R measurable, the Z-component of the JBSDE (2.35) vanishes by independence of W and J. Therefore π u (X) = Y holds, for (Y, U ) solution to the JBSDE T

Z

(Ks2

Yt = G(JT ) +

2 1/2

− |ξs | )



t

X

ij 2 λij s (Us )

i,j∈I j6=i

 12

Z

T

ds − t

X

esij . Usij dN

(2.36)

i,j∈I j6=i

Now the results of [CS12] imply that the solution (Y, U ) to (2.36) is Markovian, i.e. Yt = u(t, Jt ) and Utij = u(t, ej ) − u(t, ei ),

i, j ∈ I,

t ∈ [0, T ],

(2.37)

for a deterministic function u : [0, T ] × Rm → R. Furthermore, u is such that the associated column vector function of time u(t) := (ui (t))i∈I ∈ Rm , with ui (t) := u(t, ei ), i ∈ I, solves the coupled system of ODEs 2 2 1/2  X ij  2  12 X ij dui = − K i (t) − ξ i (t) − α (t) uj (t) − ui (t) , (2.38) α (t) uj (t) − ui (t) dt j∈I

j∈I

i ∈ I, with terminal condition u(T ) = (G(ei ))i∈I , where we use the notation K i (t) := K(t, ei ), ξ i (t) := ξ(t, ei ), i ∈ I, t ∈ [0, T ]. Hence the good-deal bound is the solution of a system of ODEs, which by reversing the time can be transformed into an initial value problem easily solved by numerical ODE solver. Example with explicit formulas for stronger Sharpe ratio constraints: Instead of considering specific conditions on contingent claims, or on the underlying jump process, let us focus on the no-good-deal constraint itself by considering a stronger Sharpe ratio constraint. Indeed the good-deal hedging strategy can be obtained explicitly using Corollary 2.21 if the Sharpe ratio constraint in the correspondence defined in (2.25) is reinforced by requiring rather √ max{kγkL2 (λt ) , |β|} ≤ Kt / 2, t ∈ [0, T ]. (2.39) Recall that the Euclidian norm | · |2 and maximum norm | · |∞ are equivalent in R2 with √ | · |∞ ≤ | · |2 ≤ 2| · |∞ . Noting this, the upper (resp. lower) good-deal bounds obtained from the constraint correspondence (2.25) can be estimated from below (resp. above) by those obtained from the stronger Sharpe ratio constraint (2.39). We generalize (2.39) by decoupling the no-good-deal constraint into kγkL2 (λt ) ≤ Ktγ

and β tr At β ≤ (Ktβ )2 , t ∈ [0, T ],

(2.40)

where A is a predictable Rn×n -matrix-valued process with symmetric values which are elliptic uniformly in (t, ω) ∈ [0, T ] × Ω, and K γ , K β are positive bounded predictable processes

Section 2.3. Case of uniformly bounded correspondences

Page 76

√ satisfying K β > ξ tr Aξ + ε for some ε > 0. Under this generalized version we obtain below a ¯ The correspondence associated to closed-form expression for a good-deal hedging strategy φ. γ β (2.40) is given for t ∈ [0, T ] by Ct = Ct × Ct with Ctγ = u ∈ L2 (λt ) : u > −1 and kukL2 (λt ) ≤ Ktγ 



and Ctβ = x ∈ Rn : xtr At x ≤ (Ktβ )2 . 



β Assuming that A−1 αt0 , where αt0 is the ellipticity t (Ker σt ) = Ker σt and that |ξt | < Kt constant of A−1 t , t ∈ [0, T ]. It follows from Corollary 2.21 and the upcoming Theorem 3.17 in Chapter 3 that the unique good-deal hedging strategy φ¯ is given by

p



φ¯t =

tr −1 ⊥ Π⊥ t (Zt ) At Πt (Zt )

(Ktβ )2 − ξttr At ξt

1/2

At ξt + Πt (Zt ),

1/2

t ∈ [0, T ],

for (Y, Z, U ) solving the JBSDE (2.18).

2.3.2

Results for ellipsoidal constraint and uncertainty about jump intensities

We are also concerned with good-deal valuation and robust hedging with respect to uncertainty about intensities of jumps in the market. Here the investor faces uncertainty about the market price of jump risk which translates into Knightian uncertainty (ambiguity) about the real world measure. We assume that from empirical/historical data, the investor has isolated a confidence region R of candidate reference measures (subjective priors) centered around a probability measure P and described by n

e)dP R := P ψ ∼ P : dP ψ = E(ψ ∗ µ

o

(2.41)

t ∈ [0, T ].

(2.42)

with ψ ≤ ψ ≤ ψ¯ ,

e functions satisfying where −1 < ψ ≤ 0 ≤ ψ¯ are P-measurable

∃K ∈ (0, ∞) s.t.

Z

 |ψ t (e)|2 + |ψ¯t (e)|2 λt (de) ≤ K,

E

Under each reference measure P ψ ∈ R, we impose an ellipsoidal no-good-deal constraint on the market price of diffusion risk and zero no-good-deal constraint on the market price of jump-risk. In other words, the no-good-deal restriction is only imposed on the β-component of the Girsanov kernels (γ, β) of pricing measures in terms of an ellipsoidal correspondence and the γ-component is set to zero. The resulting set Qngd (P ψ ) ⊆ Me (S, P ψ ) = Me (S, P ) =: Me of no-good-deal measures under P ψ is n

o

Qngd (P ψ ) := Qβ ∼ P ψ : dQβ = E(β · W )dP ψ , β ∈ C β , β ∈ −ξ + Ker σ , n

o

(2.43)

where Ctβ (ω) = x ∈ Rn : xtr At (ω)x ≤ (Ktβ )2 , (t, ω) ∈ [0, T ]×Ω, with A being a predictable Rn×n -matrix-valued process with symmetric values that are elliptic and bounded (in operator

Section 2.3. Case of uniformly bounded correspondences

Page 77

norm) uniformly in (t, ω), and K β is a positive bounded predictable process satisfying K β > √ tr ξ Aξ + ε for some ε > 0. The radial case corresponds to A ≡ IdRn . As in Section 3.2.1 of βp 0 Chapter 3, assume the separability condition A−1 αt , t (Ker σt ) = Ker σt and that |ξt | < Kt where αt0 is the ellipticity constant of A−1 , t ∈ [0, T ]. In Chapter 3 we will deal with robustness t with respect to uncertainty about the drift of traded assets in a Brownian setting, following a worst-case multi-prior approach to ambiguity as in [GS89, CE02]. Here we consider a similar approach for uncertainty about the intensity of the underlying jumps described by the priors P ψ ∈ R. A seller who seeks for robustness can charge the largest valuation bound over all priors, in order to compensate for the eventual misspecification of intensities of the jumps. In this respect, for contingent claims X ∈ L2 , the worst-case approach under uncertainty yields the good-deal bounds πtu (X) = ess sup EtQ [X], t ∈ [0, T ]. (2.44) Q∈Qngd

where Qngd :=

ngd (P ψ ). ψ≤ψ≤ψ¯ Q

S

Clearly, one can rewrite

πtu (X) = ess sup ess sup EtQ [X]. ψ≤ψ≤ψ¯ Q∈Qngd (P ψ )

By Yor’s formula, it is seen that n

o

(2.45)

Qngd = Qψ,β ∼ P : (ψ, β) ∈ C, β ∈ −ξ + Ker σ , where C = C γ × C β with n

o

Ctγ (ω) := ψ ∈ L2 (λt (ω)) : ψ t (ω) ≤ ψ ≤ ψ¯t (ω) ⊆ L2 (λt (ω)), (t, ω) ∈ [0, T ] × Ω. Hence Qngd is m-stable and convex (cf. Lemma 2.11). By (2.42), uniform ellipticity of A and boundedness of K β , the correspondence C satisfies Assumption 2.13. Moreover by standard measurable selection arguments, the associated closed-convex-valued correspondence Ce defined in (2.11) clearly satisfies Assumption 2.9. Hence the set Qngd falls in the general framework of Section 2.2 for a set of no-good-deal measure defined as in (2.12) with the associated correspondence C = C γ × C β satisfying the uniform boundedness and measurability hypotheses of Theorem 2.16. Note that the main difference between the two constraints is that Ctγ (ω) from (2.40) is given in terms of a L2 -bound on the integrands γ of the Girsanov kernels, whereas the current one is given in terms of pointwise bounds on the integrands γ. For Z ∈ H2 and U ∈ Hν2 , the optimal Girsanov kernels (¯ γ , η¯) of part a) of Lemma 2.14, can be explicitly derived from the corresponding maximization problem (2.17) and for t ∈ [0, T ] as 

γ¯t = ψ t 1{Ut 0} + 01{Ut =0}

and η¯t = 

Ktβ

2

− ξttr At ξt

1/2

tr −1 ⊥ Π⊥ t (Zt ) At Πt (Zt )

−1 ⊥ 1/2 At Πt (Zt ),

Section 2.4. Case of non-uniformly bounded correspondences

Page 78

with γ¯ clearly satisfying γ¯ > −1. Part a) of Theorem 2.16 now applies and the good-deal ¯ bound π·u (X), for X ∈ L2 , is described by πtu (X) = Yt = EtQ [X], t ∈ [0, T ], for the worst-case ¯ = Qγ¯,−ξ+¯η in Qngd and (Y, Z, U ) ∈ S 2 × H2 × Hν2 solution to the no-good-deal measure Q Lipschitz JBSDE (2.18) which in the present setup rewrites explicitly as T

Z

Yt = X + t



− ξstr Πs (Zs ) + +

T

Z t

Zstr dWs −

Ksβ

2

− ξstr As ξs

1/2 

tr −1 ⊥ Π⊥ s (Zs ) As Πs (Zs )

Z  Z t

1/2 



ψ s (e)1{Us (e)0} Us (e)λs (de) ds

E





T

(2.46)

Z E

e(ds, de). Us (e)µ

To show that the correspondence C = C γ × C β satisfies (2.22), note at first that since A is uniformly bounded in the operator norm, there exists a constant a ∈ (0, ∞) such that √ kAt (ω)k ≤ a for a.a. (t, ω). Now if |x + ξt | <  holds, then by the inequality K β > ξ tr Aξ + ε one has 1/2

xtr At x

≤ (x+ξt )tr At (x+ξt )

1/2

+ ξttr At ξt

1/2

< kAt k1/2 |x+ξt |−ε+Ktβ < a1/2 −ε+Ktβ .

Now choosing  ∈ (0, 1) such that  ≤ εa−1/2 implies that (2.22) holds. Hence the correspondence C satisfies the conditions of Corollary 2.21, which together with the results of Section ¯ 3.2.1 in Chapter 3 (cf. Theorem 3.17 therein) implies that the good-deal hedging strategy φ(X) is uniquely given by 

φ¯t (X) =

1/2

tr −1 ⊥ Π⊥ t (Zt ) At Πt (Zt )

(Ktβ )2 − ξttr At ξt

1/2

At ξt + Πt (Zt ),

t ∈ [0, T ],

for (Y, Z, U ) being solution to the JBSDE (2.46). Note that since P ngd = ψ≤ψ≤ψ¯ P ngd (P ψ ) then, as expected, the good-deal hedging strategy φ¯ is also robust with respect to uncertainty in S

¯

the sense that its tracking error Rφ (X) satisfies a supermartingale property under all measures in P ngd (P ψ ) uniformly for all reference priors P ψ ∈ R. For similar results in the Brownian setting, we refer to Chapter 3 with uncertainty about market price of (diffusion) risk and to Chapter 4 with uncertainty about the volatility of tradeable assets.

2.4

Case of non-uniformly bounded correspondences

Beyond Assumption 2.13, let us now consider a convex-valued correspondence C that still satisfies (2.10) but may fail to be uniformly bounded. In this case the generator of the JBSDE (2.18) may not be Lipschitz continuous, and results on Lipschitz JBSDEs may not apply as in the case of uniformly bounded correspondences. However for a non-uniformly

Section 2.4. Case of non-uniformly bounded correspondences

Page 79

bounded correspondence C, one can still derive approximations of the good-deal bound π·u (X) by solutions to Lipschitz JBSDEs arising from truncations of the correspondence C which satisfy Assumption 2.13. Here the density processes Γ of no-good-deal measures may not be in S 2 and X ∈ L2 may no longer imply X ∈ L1 (Q) for all Q ∈ Qngd . For this reason, we shall restrict the study here to financial claims X ∈ L∞ . We consider a  sequence Ctk := (γ, β) ∈ Ct : kγk2L2 (λt ) + |β|2 ≤ k 2 , k ∈ N, of correspondences satisfying Assumption 2.13. Since C is convex-valued and satisfies (2.10), then each C k is also convexvalued and satisfies (2.10). Moreover, since the correspondence Ce given as in (2.11) satisfies Assumption 2.9, one can show using arguments similar to those in the proof of Lemma 2.22 n o  −1/2 k k 2 n e ¯ that the correspondences Ct = (γ, β) ∈ L (λ) × R : ζt 1{ζt >0} γ, β ∈ Ct are also k k P-measurable, where C¯t (ω) denotes the closure of Ct (ω) in L2 (λt (ω)) × Rn . Since each C k , k ∈ N, satisfies Assumption 2.13, then results of Section 2.3 are applicable if one replaces C by any of the C k , k ∈ N. In addition Ctk (ω) % Ct (ω), as k % ∞. For k ∈ N, denote Qngd the set defined in (2.12) with C k instead of C and consider the associated process k πtu,k (X) = ess sup EtQ [X], t ∈ [0, T ]. Q∈Qngd k

The correspondences C k can be interpreted as describing a no-good-deal constraint consisting of the initial constraint in C in addition to a constraint on the instantaneous Sharpe ratios given by the constant bound K = k ∈ N. Note that the sets Qngd k , k ∈ N, also are convex and m-stable (by Lemma 2.11). First we have the following Lemma 2.25. Let X ∈ L∞ . Then the following dynamic principles hold: 1. π u (X) is the smallest adapted c`adl`ag process such that it is a supermartingale under every Q ∈ Qngd with terminal value X. 2. For all k ∈ N, π u,k (X) is the smallest adapted c`adl`ag process such that it is a supermartingale under every Q ∈ Qngd with terminal condition X. k Proof. The supermartingale properties of π u (X) and π u,k (X) respectively under Q ∈ Qngd and Q ∈ Qngd are consequences of m-stability and convexity of Qngd and Qngd (see Lemma k k 2.10). That they are respectively the smallest ones follows from definitions as essential suprema of closed martingales of the type E·Q [X]. For non-uniformly bounded correspondences, the following Theorem 2.26 describes in detail the approximation of good-deal valuation bounds for unbounded correspondences C by solutions to JBSDEs obtained from truncations C k of C, which are uniformly bounded and fit to the setting of Section 2.3. This is an analogue of Theorem 3.7 for a possibly discontinuous filtration. Note however the presence of an additional part (part 5.) here. We mention that both theorems

Section 2.4. Case of non-uniformly bounded correspondences

Page 80

are only concerned with approximations for good-deal valuation bounds, and not with hedging strategies. It is an interesting question if the hedging strategies associated to the approximating bounds converge in some sense to a process that can somehow be interpreted as hedging strategy. We do not investigate further on this issue. The proof of Theorem 2.26 is postponed to Appendix 2.5. Theorem 2.26. Let C be a correspondence satisfying (2.10) and such that Assumption 2.9 holds. For a contingent claim X ∈ L∞ , hold: 1. πtu,k (X) % πtu (X) a.s. as k → ∞, for all t ∈ [0, T ] 2. For any k ∈ N, π u,k (X) = Y k for (Y k , Z k , U k ) ∈ S ∞ × H2 × Hν2 unique solution to the Lipschitz JBSDE Z

T

Yt = X +





t



Z

T

t

ξstr Πs (Zs )

Zstr dWs −

Z

T

t

ess sup

+

¯ k +(0,ξs ) (γs ,ηs )∈C s ηs ∈Ker σs

(ηstr Π⊥ s (Zs )

Z



Us (e)γs (e)λs (de)) ds

+ E

Z E

(2.47)

e(ds, de), Us (e)µ

b the Doob-Meyer decompositions 3. π u (X) and π u,k (X) for k ≥ kξk∞ admit under Q c +U ∗µ e − A, and π u (X) = π0u (X) + Z · W

(2.48)

c + Uk ∗ µ e − Ak , π u,k (X) = π0u,k (X) + Z k · W

(2.49)

b × H2 (Q) b and A, Ak are non-decreasing predictable processes where (Z, U ) ∈ H2 (Q) ν b Ak = A0 = 0, and with AT , AkT ∈ L2 (Q), 0 k

A =

Z

·

ess sup ¯ k +(0,ξt ) 0 (γt ,ηt )∈C t ηt ∈Ker σt



k ηttr Π⊥ t (Zt )

Z

+ E



Utk (e)γt (e)λt (de) dt.

(2.50)

b Fu ), Z k converges to Z weakly 4. For all u ∈ [0, T ], Aku converges to Au weakly in L2 (Ω, Q, 2 k b ⊗ dt), and U converges to U weakly in L2 (Ω × E × [0, u], P ⊗ ν) in L (Ω × [0, u], Q as k → ∞. ¯

¯ in the L1 -closure of Qngd , such that π u (X) = E Q [X], then π·u (X) is a 5. If there exists Q 0 b for any ¯ quasi-left-continuous Q-martingale, and π·u,k (X) converges to π·u (X) in S p (Q)  k  1/2 c + (U k − U ) ∗ µ b and e T converges to 0 in L1 (Q), p ∈ [1, ∞). Moreover (Z − Z) · W b b with E Q Ak converges to A in S 1 (Q) [AT ] ≤ 2kXk∞ , with Z k , Z, U k , U from 3..

¯ for π u (X) in In Theorem 2.26, part 5., the hypothesis on existence of a worst-case measure Q · the L1 -closure of Qngd is ensured for any contingent claim X ∈ L∞ if the set of densities ZTQ

Section 2.4. Case of non-uniformly bounded correspondences

Page 81

(with respect to P ) of measures Q in Qngd is weakly relatively compact in L1 (i.e. uniformly integrable, by Dunford-Pettis compactness theorem [DM78, Chapter II, Theorem 25]). This is a consequence of James’ theorem (cf. [AB06, Theorem 6.36]). In case the correspondence C is ¯ is proved in part a) of Theorem 2.16, and the approximations uniformly bounded, existence of Q in Theorem 2.26 are not necessary in the first place. An example of a correspondence that may not satisfy Assumption 2.13 and for which the set of densities of measures in Qngd is uniformly integrable is given by n

Ct = (γ, β) ∈ L2 (λt ) × Rn : γ > −1

and

1 2 |β| + 2

Z E

o

g˜ 1 + γ(e) λt (de) ≤ K , (2.51) 

with K ∈ (0, ∞) and the nonnegative function g˜ defined by g˜(y) = y log y − y + 1, y > 0. Such a correspondence results from a no-good-deal constraint imposed as a dynamic bound   K 2 (σ − τ ) on the conditional relative entropy Eτ ΓΓστ log ΓΓστ for stopping times τ ≤ σ ≤ T and density processes Γ from Qngd (see e.g. also [Kl¨o06, Chapter 3]). For the correspondence C, uniform integrability of Qngd is ensured by applying the de la Vall´ee Poussin’s theorem [DM78, ¯ to Chapter II, Theorem 22]. In lack of more general assumptions for the worst-case measure Q exists, its existence could be checked in some specific situations, using the specific structure of the claim X in the model at hand. An example for this will be given in Section 3.2.2 of Chapter 3, where X is a put option in the Heston model, F is the augmented Brownian filtration and C is a radial correspondence modeling an unbounded constraint on the instantaneous Sharpe ratios (as e.g. in Section 2.4.1 below).

2.4.1

Results for constraint on instantaneous Sharpe ratios (unbounded case)

Consider good-deal bounds from a constraint on a Sharpe ratio described by the radial correspondence n

o

Ct = (γ, β) ∈ L2 (λt ) × Rn : γ > −1, kγk2L2 (λt ) + |β|2 ≤ Kt2 ,

(2.52)

for a positive predictable process K that could unbounded. Similarly to Section 2.3.1, C is convex-valued, satisfies (2.10), and is such that Assumption 2.9 holds for the associated correspondence Ce defined as in (2.11). However C does not satisfy Assumption 2.13 if K is unbounded, since for βt := K|β 0 |−1 β 0 , with β 0 ∈ Rn \ {0}, one has (0, β) ∈ C but sup(t,ω) |βt (ω)| ≥ |K|∞ = ∞. Hence the correspondence C in (2.52) fits the setup of Section 2.4 and therefore the associated good-deal bounds can be described by Theorem 2.26.

2.4.2

Results for constraint on optimal expected growth rates

For another application, we consider good-deal bounds emanating from a constraint on the optimal expected growth rates of log-returns. The set Qngd for such a constraint can be

Section 2.4. Case of non-uniformly bounded correspondences

Page 82

formulated in terms of a bound on the conditional reverse relative entropy of no-good-deal measures with respect to the reference measure P . Recall that for stopping times τ ≤ σ and a measure Q equivalent to P with density process Γ, the Fτ -conditional reverse relative entropy Hτσ (P | Q) of Q with respect to P is the conditional f -divergence (for f = − log) defined as   Eτ − log ΓΓστ =: Hτσ (P | Q) ≥ 0, with non-negativeness following from the P -submartingale property of − log Γ (cf. Proposition 2.27). [Kl¨o06, KS07b] studied dual representations of (static) good-deal bounds and their dynamic properties in a L´evy framework with constraints on the f -divergence for diverse choices of the function f corresponding to logarithmic (f (z) = − log(z)), exponential (f (z) = z log z corresponding to constraint (2.51) on the conditional relative entropy) and power (f (z) = z p , p ≥ 1) utility functions. Related pricing (and hedging) approaches from a constraint on generalized relative entropy are considered in [Lei08]. In [Bec09] it was shown in a Brownian setting that a dynamic bound on the reverse relative entropy of risk-neutral measures (in Me ) corresponds to a bound on the optimal expected growth rates of (log-)returns, for any extension of the financial market by additional derivative price processes that are computed as conditional expectations under no-good-deal pricing measures (in Qngd ). This provides a no-good-deal constraint that, in the Brownian setting, is essentially equivalent to the constraint on the instantaneous Sharpe ratios. We first note that for a discontinuous filtration (presence of jumps), the two no-good-deal constraints are no longer equivalent and the constraint on the Sharpe ratios as in Section 2.3.1 appears mathematically more tractable in terms of JBSDEs. In fact, the correspondence resulting from a constraint on optimal growth rates may fail to satisfy Assumption 2.13, even when the growth rates are bounded by a constant. In this section, we derive good-deal bounds and show existence of hedging strategies for a no-good-deal constraint on optimal growth rates, using Lipschitz JBSDEs in a setup with jumps of finite state semi-Markov processes, i.e. in particular having a finitely-supported jump compensator. The restriction to finite state space is important as this ensures that the resulting JBSDEs have classical Lipschitz continuous generators and existence of good-deal hedging strategies can be shown as in Theorem 2.19. Beyond finitely supported compensators, the results in this section may not guarantee existence of a good-deal hedging strategy for such no-good-deal constraints since the associated correspondence C in (2.57) may be unbounded; yet we do still have result on good-deal valuation bounds. To be more precise in our current setup, let K be a positive bounded predictable process and define Qngd as consisting of measures Q ∈ Me that satisfy Hτσ (P

1  | Q) ≤ Eτ 2

Z τ

σ

Ku2 du , 

for all τ ≤ σ ≤ T,

(2.53)

where τ, σ are stopping times. Let us recall [Bec09, Proposition 2.2] providing some useful properties of the process − log Γ, for Q ∼ P with finite reverse entropy. Proposition 2.27. Let Q ∼ P with density process Γ of Q with respect to P such that log ΓT ∈

Section 2.4. Case of non-uniformly bounded correspondences

Page 83

L1 . Then − log(Γ) is a P -submartingale of class (D) with a Doob-Meyer decomposition − log(Γ) = N + A, where N is a uniformly integrable P -martingale and A a predictable,   non-decreasing and P -integrable process, with N0 = A0 = 0. Moreover Eτ − log ΓΓστ = Eτ [Aσ − Aτ ] holds for all stopping times τ ≤ σ ≤ T . Using Proposition 2.27, we can reformulate the definition of Qngd in terms of deterministic Q times. To this end, measures υh and κ = κ i on the predictable σ-field P hR define positive i RT T 1 2 by υ(B) := 2 E 0 1B Ku du and κ(B) := E 0 1B dAu , B ∈ P, where A is the nondecreasing process in the Doob-Meyer decomposition of − log(Γ), for Γ being the density process of a measure Q ∼ P . We have the following equivalent condition for Q ∼ P to be in Qngd (see [Bec09, Proposition 2.3]). Proposition 2.28. For Q ∼ P with density process Γ satisfying 

Es − log

Γt  1  ≤ Es Γs 2

Z s

t

Ku2 du for all deterministic times s ≤ t ≤ T, 

(2.54)

holds κ(B) ≤ υ(B) for any B ∈ P. In particular, condition (2.54) is equivalent to its stopping time analogue (2.53). Thus a measure Q ∈ Me is element of Qngd if and only if (2.54) holds. One can interpret, as mentioned above, the constraint (2.53) as a bound on the optimal expected growth rates in the financial market extended by additional derivative price processes (see [Bec09, Theorem 3.1]). Using Lemma 2.8, one can formulate a definition of the set Qngd by a condition on the Girsanov kernels of the associated measures. For Q ∈ Me , the following proposition derives N and A from Proposition 2.27 in our setup in terms of the Girsanov kernels of Q. The proof is deferred to Appendix 2.5. Proposition 2.29. Let Q ∼ P with Girsanov kernels (γ, β) and density process Γ = E(M ) e as in part b) of Lemma 2.1, and let also g be defined by (2.3). Then where M = β · W + γ ∗ µ ⊥ 1. If Qγ,β ∈ Me with E[− log Γγ,β T ] < ∞, then β = −ξ + η, with Πt (βt ) = ηt , t ∈ [0, T ]. γ,β γ,β γ,β Moreover the Doob-Meyer decomposition − log Γ = N +A is given by

e and N γ,β = −β · W − log(1 + γ) ∗ µ Z · Z  1 |βs |2 + g(1 + γs (e))λs (de) ds. Aγ,β = 2 0 E

(2.55)

2. If Qγ,β ∈ Qngd , then β = −ξ + η and 1 |ηt |2 + 2

1 1 g(1 + γt (e))λt (de) ≤ Kt2 − |ξt |2 , 2 2 E

Z

t ∈ [0, T ].

(2.56)

Section 2.4. Case of non-uniformly bounded correspondences

Page 84

e e-integrable function γ > −1 and any predictable µ 3. Reciprocally, any P-measurable R process β with Πt (βt ) = −ξt , t ∈ [0, T ] satisfying 12 |βt |2 + E g(1 + γt (e))λt (de) ≤ 1 2 γ,β ∈ Qngd with Girsanov kernels (γ, −ξ + η), where 2 Kt , t ∈ [0, T ], define a measure Q ⊥ ηt = Πt (βt ), t ∈ [0, T ]. b ∈ Qngd 6= ∅. From Proposition Assume that K > |ξ| + ε, for some ε ∈ (0, 1), ensuring that Q 2.29, the constraint correspondence satisfying (2.10) for the set of no-good-deal measures defined by (2.12) can be chosen as n

Ct := (γ, β) ∈ L2 (λt ) × Rn : γ > −1,

o

k2g(1 + γ)kL1 (λt ) + |β|2 ≤ Kt2 .

(2.57)

The correspondence C has non-empty convex values (since (0, 0) ∈ C and g is convex). However, it is easily seen that C given by (2.57) does not satisfy Assumption 2.13 in general. Indeed, assume that µ is the random measure of the big jumps of a Gamma L´evy process with parameters a = b = 1, i.e. with E = R \ {0}, ζ ≡ 1 and λ(dx) = exp(−x)x−1 1{x≥1} dx.  R Then for the function γ with γ(x) = exp(x/2)1{x≥1} satisfies E g 1 + γ(x) λ(dx) < ∞ and R 2 n E |γ(x)| λ(dx) = ∞. Therefore for suitably chosen K ∈ (0, ∞) the sequence ((γ , 0))n∈N of n 2 Girsanov kernels with γ = γ1[1,n] is included in C but is not bounded in L (λ). It is for this reason that we present this case as an application of Section 2.4 which works generally beyond Assumption 2.13. Note that around −1 < γ ≤ 1, one has the following Taylor approximation of g up to the 2 (leading) second order: g(1 + γ) = − log(1 + γ) + γ = γ2 + O(γ 3 ). In this sense, the constraint on Sharpe ratios can be viewed as an approximation of that on optimal growth rates, for pricing measures Qγ,β possessing a low market prices of jump-risk γ. It is therefore not surprising that for continuous filtrations (absence of jump-risk, i.e. trivial µ = ν = 0), formally γ = 0 and the two types of no-good-deal constraints are equivalent; cf. [Bec09]. Clearly, a (bounded) constraint on the Sharpe ratios is mathematically more tractable since it naturally leads to standard Lipschitz JBSDEs for good-deal valuation and hedging. The correspondence C in (2.57) has been defined so that it satisfies (2.10) but, for our theory, we will show that its associated correspondence Ce satisfies Assumption 2.9. First we describe the closure C¯t of the set Ct in L2 (λt ) × Rn . This needs some preparation because the function g and its derivative g 0 explode in the neighborhood of 0. Consider the pointwise approximation (g l )l∈N of g consisting of non-negative Lipschitz functions ( l

g (y) :=

g( 1l ), g(y),

if 0 ≤ y ≤ if y ≥ 1l .

1 l

The sequence (g l )l is non-decreasing and converges pointwise to g on (0, ∞) as l tends to infinity. In particular for any l ∈ N the function g l satisfies g l (1 + y) ≤ Const |y|2 for all y ≥ −1 for some Const > 0. This property will be useful later in the proof of the second claim of

Section 2.4. Case of non-uniformly bounded correspondences

Page 85

Lemma 2.30. Note that the function g(1 + ·) is dominated by Const |y|2 only locally around the origin. Now define for each l ∈ N the correspondence n

o

C¯tl := (γ, β) ∈ L2 (λt ) × Rn : γ ≥ −1, k2g l (1 + γ)kL1 (λt ) + |β|2 ≤ Kt2 . Since g l is continuous and non-negative, then by Fatou’s lemma the sets C¯tl are closed in L2 (λt ) × Rn , for each l ∈ N, t ∈ [0, T ]. The related correspondence Ce l according to (2.11) therefore writes n

1/2

Cetl = (γ, β) ∈ L2 (λ) × Rn : γ ≥ −ζt

1

− , 2g l (1 + ζt 2 γ) L1 (λt ) + |β|2 ≤ Kt2 .



o

T For any l ∈ N since g l ≤ g, then Ct ⊆ C¯tl , which implies C¯t ⊆ C¯tl . Hence Cet ⊆ l∈N Cetl , t ∈ [0, T ]. In fact equality holds in the latter inclusion as claimed by the following lemma, which also infers that Ce satisfies Assumption 2.9. The proof is provided in Appendix 2.5.

Lemma 2.30. For C defined in (2.57), holds Cet = closed-valued correspondence Ce is P-measurable.

el l∈N Ct ,

T

t ∈ [0, T ]. In particular the

Overall, the correspondence C defined in (2.57) and describing a no-good-deal constraint as a bound on the optimal expected growth rates in the financial market satisfies the hypotheses of Theorem 2.26. This yields an approximation of the associated good-deal bound π·u (X) in terms of solutions to Lipschitz JBSDEs for abstract random measures µ and contingent claims X ∈ L∞ . Although the correspondence C in (2.57) might not satisfy Assumption 2.13 for general random measures µ, this assumption apparently holds when the measures λt (with ν(dt, de) = λt (de)dt respect to which the compensator ν is absolutely continuous) are finitely supported. In this case the results of Section 2.3 (on valuation and hedging) are again applicable, and the good-deal bounds can be directly described as solutions to JBSDEs. Without loss of generality, we elaborate on this by considering the semi-Markov setup of Section 2.3.1. Example for semi-Markov jump-dynamics: Consider again the framework of Section 2.3.1, with a semi-Markov process J on finite state space E = {e1 , . . . , em } ⊂ Rm , and denote P ij of the jumps, compensator I = {1, . . . , m}, and counting process N = i,j∈I, j6=i N P ij ij ij ν(dt) := i,j∈I, j6=i λij t dt, for jump intensities λt = 1{Jt− =ei } αt (τt− ) with α ≥ 0 defined in (2.33) and the time the process has spent at state Jt being τt := sup{s ∈ [0, t] : Jt−u = Jt , u ∈ [t − s, t]}. The constraint in the optimization problem in Lemma 2.14 for C in (2.57) is finite dimensional and given for t ∈ [0, T ] by the set of (γ, η) ∈ (−1, ∞)m×m−1 × Rn satisfying X 1 2 1 2 2 |η| + g(1 + γ ij )λij t ≤ (Kt − |ξt | ), 2 2 i,j∈I, j6=i

(2.58)

Section 2.4. Case of non-uniformly bounded correspondences

Page 86

for g given by (2.3). We can assume without loss of generality γ ij = 0 on the set {λij = 0}. This together with (2.58) imply that g(1 + γ ij ) ≤

Kt2 − |ξt |2 2λij t

for all i, j ∈ I, j 6= i.

1{λij 6=0} t

(2.59)

Since g is continuous and limx&−1 g(1 + x) = limx→∞ g(1 + x) = +∞, then (2.59) yields compactness in (−1, ∞)m×m−1 × Rn of the set of values of ((γ ij )i,j∈I, j6=i , η) satisfying (2.58). Furthermore for Z ∈ H2 and U ∈ Hν2 , the objective function F (t, γ, η) := η tr Π⊥ t (Zt ) + P ij ij ij m×m−1 n × R and predictable in (t, ω) ∈ i,j∈I, j6=i Ut γ λt is continuous in (γ, η) ∈ (−1, ∞) [0, T ] × Ω. Hence by the usual direct method of variational analysis (cf. [ET99]) and standard measurable selection theorems (cf. [Roc76], which do not require completeness of the measure space for correspondences with finite dimensional ranges), there exists a predictable (t, ω)-wise maximizer (¯ γ , η¯) := (¯ γ (Z, U ), η¯(Z, U )) ∈ (−1, ∞)m×m−1 × Rn of F over the constraint set described by (2.58). Since by part 3. of Lemma 2.32 the function g satisfies |x| − 2 ≤ (g(1 + x))2 , for all x > −1, then (2.59) implies that γ ij ≤



(Kt2 −|ξt |2 )2 1{λij 6=0} 2 4(λij t t )

+ 2. This in turn yields (after squaring and

summing over all states ei , ej ∈ E, j 6= i) 2 ij ij γ λ t ≤

X i,j∈I, j6=i



 (K 2 − |ξ |2 )2 t t

X i,j∈I, j6=i  |K|4 ∞

4

2

1{λij 6=0} + 2 λij t

2 4(λij t )

∨2



X

t



i,j∈I, j6=i

−2 (λij t ) 1{λij t 6=0}

+1

2

(2.60) λij t ,

with the convention that 0/0 = 0. Now assume that

∃ c¯λ ≥ cλ > 0 s.t. cλ 1{λij 6=0} ≤ λij ≤ c¯λ

for all i, j ∈ I, j 6= i.

(2.61)

Condition (2.61) ensures by (2.60) that the correspondence C defined in (2.57) satisfies the uniform boundedness Assumption 2.13 in the current semi-Markov jump setup with 2

|β| +

|γ ij |2 λij t i,j∈I, j6=i X



|K|2∞

 |K|4

+ m(m − 1)



4

 1

∨2

c2λ

2

+ 1 c¯λ ,

(2.62)

for all (γ, β) ∈ C. Note that (2.61) does not exclude the fact that the intensities λij can vanish on a non-negligible set. Indeed we only require on the set where they do not vanish, that they are bounded from below by a positive constant, uniformly over all states ei , ej ∈ E and (t, ω) ∈ [0, T ] × Ω. That the intensities of the jumps are bounded from above appears as a non-restrictive assumption for practical examples, which could be interpreted as a sufficient condition preventing the rate of state-change of the semi-Markov process from exploding.

Section 2.5. Appendix

Page 87

Former calculations suggest that in the limit as m → ∞, the right-hand side of in (2.62) would tend to infinity. In the limiting case, therefore, the correspondence C may no longer be uniformly bounded; this shows the importance of restricting to finitely many states. Part a) of Theorem 2.16 applies and yields that the good-deal bound π·u (X) for X ∈ L2 is ¯ ¯ = Qγ¯,−ξ+¯η and (Y, Z, U ) ∈ S 2 × H2 × Hν2 solution to given by π·u (X) = E·Q [X] = Y for Q the BSDE (2.18) which in the present setup rewrites T

Z

Yt = X +



t





Usij γ¯sij λij s ds

i,j∈I, j6=i T

Z

X

(−ξs + η¯s )tr Zs +

t

Zstr dWs

X



Z

i,j∈I, j6=i t

T

(2.63)

e ij . Usij dN s

¯ = Qγ¯,−ξ+¯η is in fact a no-good-deal measure, i.e. In addition, the worst-case measure Q ¯ ∈ Qngd , because the optimal Girsanov kernels (¯ Q γ , −ξ + η¯) ∈ C satisfies γ¯ ij > −1 for all i, j ∈ I, j 6= i. It is also possible to obtain a qualitative result about good-deal hedging in this setting. Indeed, condition (2.22) is clearly satisfied for  = ε ∈ (0, 1) since by assumption K > |ξ| + ε. Hence applying Theorem 2.19 yields in particular existence of a good-deal hedging ¯ strategy φ¯ = φ(X) (for X ∈ L2 ) with ¯

f φ (t, Zt , Ut ) = ess inf f φ (t, Zt , Ut ),

(2.64)

φ∈Φ

for (Y, Z, U ) solution to the BSDE (2.63) and 

f φ (t, Zt , Ut ) = −ξttr φt + ess sup β tr (Zt − φt ) + (γ,β)

X



Utij γ ij λij t ,

i,j∈I, j6=i

where the supremum is taken over all (γ, β) = ((γ ij )i,j∈I, j6=i , β) ∈ (−1, ∞)m×m−1 × Rn satisfying (2.58). One could not expect to obtain for (2.64), in the generality of the present example, an explicit formula for the good-deal hedging strategy φ¯ solving the minimization problem in (2.64). Yet, approximations might be computed using numerical algorithms for convex optimization problems (cf. e.g. [BV04]), and of Lipschitz JBSDEs (cf. e.g. [BE08] for related but different types of generators).

2.5

Appendix

This appendix collects some proofs and statements of results that were omitted in the main body of the chapter. The order of appearance here is the same as in the main text. ˜ with P -characteristics Proof of Lemma 2.1. For part a), apply [JS03, Theorem III.3.24] to X P ˜ has a canonical (B, c, ν) := (0, I, ν ) with respect to the truncation function h. Note that X

Section 2.5. Appendix

Page 88

˜ = W + (IdE − h) ∗ µ + h ∗ µ e in terms of the truncation function h, where representation X IdE is the identity function on E. Part b) is a consequence of part a), the weak predictable e) with respect to (P, F), and [JS03, Proposition III.5.10, representation property (2.2) of (W, µ Theorem III.5.19 and Corollary III.5.22] which apply with Y := 1 + γ, and a = 0, Yˆ = 0 since ν  λ ⊗ dt. Proposition 2.31 ([LM78], Theorem II.5). Let M be a quasi-left-continuous local martingale satisfying ∆M ≥ −1 and define T¯ := inf {t : ∆Mt = −1} ∧ T . If the predictable compensator Λ of the process X

D = hM c i·∧T¯ +

∆Ms2 1

s≤·∧T¯

is bounded, then E martingale.

h

[E(M )]T

1/2 i

|∆Ms |≤1

+ ∆Ms 1

∆Ms >1

(2.65)



< ∞. In particular E(M ) is a uniformly integrable

We have the following lemma. Being purely analytical, the proof is omitted. Lemma 2.32. For any y ≥ 0, hold 1. (1 − 2. (1 −

√ 2 y) ≤ (y − 1)2 1{y≤2} + |y − 1|1{y>2} ≤ √

1 √ √ (1 − y)2 , 2 (1 − 2)

y)2 ≤ g(y), for the function g defined in (2.3),

3. |y − 1| − 2 ≤ (g(y))2 . Proof of Proposition 2.3. By Lemma 2.32 it follows that 0≤

Z E

1−

q

2

1 + γt (e) λt (de) ≤

Z



g 1 + γt (e) λt (de) ≤ K, t ∈ [0, T ].

E

2 √ Hence the process 1 − 1 + γ ∗ ν is locally P -integrable. Then applying [JS03, Theorem e-integrability of γ, with II.1.33, d)] (with a = 0 and γb = 0 since ν  λ ⊗ dt holds) yields the µ e being a purely discontinuous local martingale. Moreover by (2.5), β · W is well-defined as a γ ∗µ e. continuous local martingale, so that M is a local martingale with M c = β · W and M d = γ ∗ µ e  e)t = γ(t, ∆Xt )1 e By definition, the jumps of M are given by ∆Mt = ∆ (γ ∗ µ , t ∈ [0, T ], ∆X 6=0 t

e is the semimartingale X e = W + (IdE − h) ∗ µ + h ∗ µ e with h(e) := e1{|e|≤1} . Now where X

since γ > −1 then ∆M > −1, and therefore E(M ) is a positive local martingale. By [JS03, Corollary II.1.19] and ν  λ ⊗ dt,  M is quasi-left-continuous. Hence the process D in (2.65)  R· 2 2 can be written as D = 0 |βs | ds + γ 1{|γ|≤1} + γ1{γ>1} ∗ µ. By [JS03, Proposition II.1.28] the predictable P -compensator Λ of D is Λ =







2 2 0 |βs | ds + γ 1{|γ|≤1} + γ1{γ>1} ∗ ν. Now

Section 2.5. Appendix

Page 89

using (2.4) and (2.5), Lemma 2.32 yields boundedness of Λ. By Proposition 2.31 this implies that Γ = E(M ) is a positive uniformly integrable martingale. In particular Γ defines a measure Q ∼ P via dQ = ΓdP . Let (β Q , γ Q ) be the actual Girsanov kernels of Q from part a) of e. Hence Lemma 2.1. By part b) of Lemma 2.1, Γ = E(M Q ) holds with M Q = β Q · W + γ Q ∗ µ Q Q E(M ) = E(M ) and by taking stochastic logarithms one obtains M = M , or equivalently e. The left hand side is a continuous local martingale whereas (β Q − β) · W = (γ Q − γ) ∗ µ the right hand side is a purely discontinuous local martingale. By orthogonality both local martingales are equal to zero. Since from (2.5) both local martingales are square integrable, then β = β Q P ⊗ dt-a.s. and γ = γ Q P ⊗ λ ⊗ dt-a.s.. i

i

Proof of Lemma 2.11. For (γ i , β i ) ∈ C and β i = −ξ + η i , η i ∈ Ker σ, i = 1, 2, let Qγ ,β be  e . in Qngd with density processes Γi with respect to P given by Γi := E (−ξ + η i ) · W + γ i ∗ µ Convexity: Let α ∈ [0, 1] and Γ = αΓ1 + (1 − α)Γ2 . Since Me is convex, then Γ ∈ Me and corresponds to a measure Qγ,β ∼ P with Girsanov kernels (γ, β = −ξ + η), where η ∈ Ker σ. Using Itˆo’s formula and convexity of the values of C one shows that (γt , βt ) = αΓ1t− 1 1 Γt− (γt , βt )

+

(1−α)Γ2t− (γt2 , βt2 ) Γt−

∈ Ct , t ∈ [0, T ]. Hence Qngd is convex.

M-stability: Let τ ≤ T be a stopping time and Γt := 1{t≤τ } Γ1t + 1{τ ≤t} Γ1τ Γ2t /Γ2τ , t ∈ [0, T ]. Since Me is m-stable, then Γ ∈ Me and corresponds to a measure Qγ,β ∼ P with Girsanov kernels (γ, β := −ξ + η), where η ∈ Ker σ. We show that (γ, β) ∈ C. It holds that  R R et − 0t 21 |βsi |2 + E g(1 + γsi (e))λs (de) ds, for i = 1, 2, log Γit = β i · Wt + (log(1 + γ i )) ∗ µ and g being the function given by (2.3). Hence 2

2

eT − log ΓT = β · WT + (log(1 + γ )) ∗ µ

T

Z

1

|βs2 |2

Z

+



g(1 + γs2 (e))λs (de) ds

2 E 1 eτ + (β − β ) · Wτ + (log(1 + γ ) − log(1 + γ 2 )) ∗ µ Z τ Z  1 12 − (|βs | − |βs2 |2 ) + (g(1 + γs1 (e)) − g(1 + γs2 (e))λs (de) ds. 2 0 E 1

0

2

Equivalently we have Z

eT − β · WT + (log(1 + γ) ∗ µ 1

T

1

2

0

1

eτ − = β · Wτ + (log(1 + γ )) ∗ µ Z

T

+ τ



Z

T

βs2 dWs + 1

τ



1

= 1B β + 1B c β −

Z 0

T

|βs2 |2 +

2 2



T

Z τ

Z E

Z E

2

|βs | +

Z



g(1 + γs (e))λs (de) ds E

τ

1

2

0

|βs1 |2

Z



+ E

g(1 + γs1 (e))λs (de) ds

e(ds, de) (log(1 + γs2 (e))µ 

g(1 + γs2 (e))λs (de) ds 

· WT + log 1 + 1B (s)γ 1 + 1B c (s)γ 2

1 1B (s)β 1 + 1B c (s)β 2 2 +

2

Z

s

s

Z E





eT ∗µ 



g 1 + 1B (s)γs1 (e) + 1B c (s)γs2 (e) λs (de) ds,

Section 2.5. Appendix

Page 90

where B = [0, τ ] = {(t, ω) : t ≤ τ (ω)} ∈ P. Thus (γ, β) = 1B (γ 1 , β 1 ) + 1B c (γ 2 , β 2 ) ∈ C since C has convex values. Proof of Lemma 2.14. Proofs for part a) and b) are analogous, so we only prove part a). Consider the equivalent (to (2.17)) maximization problem (γt∗ (ω), ηt∗ (ω)) =

argmax ηttr (ω)Π⊥ (t,ω) (Zt (ω)) +

(γt (ω),ηt (ω))

Z

Ut (ω, e)γt (ω, e)(ζt (ω, e))1/2 λ(de),

(2.66)

E

where the maximum is taken over (γt (ω), ηt (ω)) ∈ Cet (ω) + (0, ξt (ω)) and ηt (ω) ∈ Ker σt (ω), with Ce given in (2.11). The maximization problem (2.66) is more convenient for measurable selection arguments since the range L2 (λ) × Rn of the associated correspondence Ce does not depend on t nor ω. The corresponding maximizers of (2.17) and (2.66) are related by  γ∗ (¯ γt , η¯t ) = √t 1{ζt >0} , ηt∗ , t ∈ [0, T ]. ζt

(2.67)

For all (t, ω) ∈ [0, T ] × Ω, the sets Cet (ω) are closed and convex in L2 (λ) × Rn since C¯t (ω) are closed and convex in L2 (λt (ω)) × Rn . In addition Cet (ω) are bounded in L2 (λ) × Rn (hence weakly compact) since by Assumption 2.13 the sets C¯t (ω) are bounded in L2 (λt (ω)) × Rn . As a   consequence Cet (ω) + (0, ξt (ω)) ∩ L2 (λ) × Ker σt (ω) is also weakly compact in L2 (λ) × Rn . Since Z ∈ H2 and U ∈ Hν2 , the objective function of the maximization problem (2.66) is linear and continuous in (γ, η) ∈ L2 (λ) × Rn , t ∈ [0, T ]. Hence by the direct method in variational analysis (see [ET99]), there exists for all (t, ω) ∈ [0, T ] × Ω a maximizer (γt∗ (ω), ηt∗ (ω)) in   Cet (ω) + (0, ξt (ω)) ∩ L2 (λ) × Ker σt (ω) . Now we show that one can choose (γ ∗ , η ∗ ) such that η ∗ is P-measurable and γ ∗ (hence γ¯ = γ ∗ ζ −1/2 1{ζ>0} ) is P ⊗ E-measurable (since ζ clearly is). Note that the Hilbert space L2 (λ) is separable (hence is a Polish space) and   also that the correspondence Ce + (0, ξ) ∩ L2 (λ) × Ker σ is P-measurable since Ce is Pmeasurable by Assumption 2.9 and ξ, σ are predictable processes. Since Z is predictable and e U is P-measurable, the objective function is P-measurable in (t, ω) ∈ [0, T ] × Ω and hence a Carath´eodory function defined on [0, T ] × Ω × L2 (λ) × Rn . By standard measurable selection [AF90, Theorems 8.1.3, 8.2.11] one obtains (γ ∗ , η ∗ ) satisfying (2.66) for all (t, ω) ∈ [0, T ] × Ω, with η ∗ P-measurable and γ ∗ P − B(L2 (λ))-measurable. Let us show that γ ∗ defined by γ ∗ (t, ω, e) := γ ∗ (t, ω)(e) is actually P ⊗ E-measurable. Denote by (un )n∈N an orthonormal P basis of L2 (λ). Then γt∗ (ω) has the decomposition γt∗ (ω) = n∈N hγt∗ (ω), un iL2 (λ) un for any (t, ω) ∈ [0, T ] × Ω. Now since for each n ∈ N the map L2 (λ) 3 γ 7→ hγ, un iL2 (λ) is continuous, then hγ ∗ , un iL2 (λ) is a P-measurable process for all n ∈ N. Thus γ ∗ is P ⊗ E-measurable as a countable sum of P ⊗ E-measurable functions. Now by approximation of measurable functions e by simple functions, one can make η ∗ predictable and γ ∗ P-measurable through modification on a P ⊗ dt-null-set. The corresponding (¯ γ , η¯) given by (2.67) then solves (2.17) for P ⊗ dt-almost all (t, ω) ∈ [0, T ] × Ω.

Section 2.5. Appendix

Page 91

Proof of Theorem 2.16. Part a) and b) are analogous, so we only prove part a). Denote by f the generator of the JBSDE (2.18). For all t ∈ [0, T ], ft is, by part a) of Lemma 2.14, the supremum over (γ, −ξ + η) ∈ C¯ and η ∈ Ker σ of a family of linear generators ftγ,η (t, z, u)

tr

Z

:= (−ξt + ηt ) z +

u(e)γt (e)λt (de), E

where coefficients (γ, (−ξ in Rn × L2 (λt (ω)) uniformly in (t, ω) by Kf :=  + η)) are bounded  sup(t,ω) sup(γ,β)∈C(t,ω) kγkL2 (λt ) + |β| ∈ (0, ∞). The generator f is then Lipschitz contin¯ n uous in (z, u) ∈ R × L2 (λt (ω)), uniformly in (t, ω) ∈ [0, T ] × Ω with Lipschitz constant Kf and satisfies ft (0, 0) = 0. By [Bec06, Proposition 3.2], the JBSDE (2.18) has a unique ¯ the process solution (Y, Z, U ) ∈ S 2 × H2 × Hν2 . Now recall that for each (γ, β) ∈ C, R 2 β is bounded and E γt (e)λt (de) is uniformly bounded in t ∈ [0, T ]. Hence by Lemma 2.4 and the subsequent remark, the JBSDEs with generators f γ,η have unique solutions γ,β (Y γ,η , Z γ,η , U γ,η ) ∈ S 2 × H2 × Hν2 , which satisfy Ytγ,η = EtQ [X], t ∈ [0, T ], for β = −ξ + η. ¯ ¯ Furthermore since f = f γ¯,¯η holds, then one also has Yt = EtQ [X], Q-a.s.. From Lemma 2.15 ¯ Q u ¯ it holds that πt (X) ≥ Et [X], Q-a.s., and so to conclude the proof, one has to show that πtu (X) ≤ Yt , P -a.s.. For all (γ, β := −ξ + η) ∈ C (defining Qγ,β ∈ Qngd ) holds ft (Zt , Ut ) = ftγ¯,¯η (Zt , Ut ) ≥ ftγ,η (Zt , Ut ),

t ∈ [0, T ],

for (Y, Z, U ) solution to the JBSDE (2.18). Moreover since f γ,η are Lipschitz in (z, u) with uniform Lipschitz constants Kf and ftγ,η (Ztγ,η , Utγ,η )



ftγ,η (Ztγ,η , Ut )

Z

= E

γt (e)(Utγ,η (e) − Ut (e))λt (de),

t ∈ [0, T ],

e) being a uniformly integrable martingale by Proposition 2.31, then with E ((−ξ + η) · W + γ ∗ µ Proposition 2.6 implies that Yt ≥ Ytγ,η , P -a.s., for all (γ, β = −ξ + η) ∈ C. As a consequence Yt ≥ ess sup(γ,η) Ytγ,η = πtu (X), P -a.s., where (γ, −ξ + η) range over all Girsanov kernels of measures Q ∈ Qngd .

Proof of Lemma 2.17. Denote by f φ (for φ ∈ Φ) the generator of the JBSDE (2.20). By part b) of Lemma 2.14, the generator f φ is the supremum of a family of affine JBSDE R (φ,γ,β) generators ft (t, z, u) = −ξttr φt + (Zt − φt )tr βt + E Ut (e)γt (e)λt (de), with coefficients (γ, β) ∈ C¯ bounded in Rn × L2 (λt (ω)) uniformly in (t, ω) ∈ [0, T ] × Ω by the constant  Kf := sup(t,ω) sup(γ,β)∈C(t,ω) kγkL2 (λt ) + |β| ∈ (0, ∞). Hence for all φ ∈ Φ, f φ is Lipschitz ¯ continuous in (z, u) ∈ Rn × L2 (λt (ω)) uniformly in (t, ω) ∈ [0, T ] × Ω with Lipschitz constant Kf , and satisfies ftφ (0, 0) ∈ H2 since ξ is bounded and φ ∈ H2 . By [Bec06, Proposition 3.2], the JBSDE (2.20) has a unique solution (Y φ , Z φ , U φ ) ∈ S 2 × H2 × Hν2 .

Section 2.5. Appendix

Page 92

c and Z e = Z − φ, so that using (2.20) gives Now let Ye := Y φ − φ · W

−dYet = − dYtφ + ξttr φt dt + φtr t dWt 

e U) + = Zettr β˜t (Z,

Z E

Z  tr e e e(dt, de). Ut (e)˜ γt (Z, U )(e)λt (de) dt − Zt dWt − Ut (e)µ E

e solves the JBSDE (2.19) with terminal value YeT = X − φ · W cT ∈ L2 . This means that (Ye , Z)   cT . Finally translation invariance Hence part b) of Theorem 2.16 implies that Yet = ρt X − φ · W 

ct = ρt X − of ρ yields Ytφ = Yet + φ · W



RT

c φtr s dWs , t ∈ [0, T ].

t

Proof of Lemma 2.18. Consider deterministic (and time-independent) parameters z ∈ Rn , u ∈ L2 (ζλ), σ ∈ Rd×n , ξ ∈ Im σ tr , and for a convex closed and bounded set C¯ ⊆ L2 (ζλ) × Rn , consider the function L : Rn × (L2 (ζλ) × Rn ) → R defined by tr

tr

(φ, (γ, β)) 7→ L(φ, (γ, β)) := −ξ φ + β (z − φ) +

Z

u(e)γ(e)ζ(e)λ(de). E

Clearly for any fixed φ ∈ Rn the function (γ, β) 7→ L(φ, (γ, β)) is linear and bounded, and for any fixed (γ, β) the function φ 7→ L(φ, (γ, β)) is linear and continuous. Since the set C¯ is convex closed and bounded, it is weakly compact in L2 (ζλ) × Rn . Now since Im σ tr is convex and closed, then by [ET99, Proposition 2.3, Chapter VI] the minimax identity inf

sup L(φ, (γ, β)) = sup

φ∈Im σ tr (γ,β)∈C ¯

inf

¯ φ∈Im σ (γ,β)∈C

tr

L(φ, (γ, β))

(2.68)

holds. Plus, the right hand side of (2.68) is equal to sup

inf

¯ φ∈Im σ (γ,β)∈C

L(φ, (γ, β)) = sup tr



Z

tr

β z+

u(e)γ(e)ζ(e)λ(de) +

¯ (γ,β)∈C

=

sup ¯ (γ,β)∈C Π(β)=−ξ

E

β tr z +

inf

φ∈Im σ



φtr (ξ + Π(β)) tr

Z

u(e)γ(e)ζ(e)λ(de), E

since inf φ∈Im σtr φtr (ξ + Π(β)) equals 0 if Π(β) = −ξ and −∞ otherwise. Now extending the arguments to random and time-dependent parameters clearly gives (2.21). Proof of Theorem 2.19. Consider deterministic (and time-independent) parameters z ∈ Rn , u ∈ L2 (ζλ), σ ∈ Rd×n , ξ ∈ Im σ tr , and for a convex bounded set C¯ ⊆ L2 (ζλ) × Rn satisfying ¯ consider the following analog of f φ as a function of φ: {0} × B (−ξ) ⊆ C, Rn ⊇ Im σ tr 3 φ 7→ F (φ) := −ξ tr φ + ess sup β tr (z − φ) + ¯ (γ,β)∈C

Z



u(e)γ(e)ζ(e)λ(de) . E

The function F is clearly convex and continuous. Moreover F is coercive on Im σ tr , i.e. ¯ one gets the F (φ) → ∞ as |φ| → ∞ for φ ∈ Im σ tr . Indeed, using {0} × B (−ξ) ⊆ C,

Section 2.5. Appendix

Page 93

estimate F (φ) ≥ −ξ tr φ + ess supβ∈B (−ξ) β tr (z − φ) = −ξ tr z +  |z − φ|, which clearly implies coercivity of F . Hence by [ET99, Chapter II, Proposition 1.2], the function F admits a minimizer in Im σ tr . By extending the arguments to random and time-dependent parameters, ¯ existence of φ¯ ∈ Φ satisfying f φ (t, Zt , Ut ) = ess infφ∈Φ f φ (t, Zt , Ut ) follows by standard measurable selection arguments. Recall by Lemma 2.18 that it holds in particular that ¯ f φ (t, Zt , Ut ) = ess infφ∈Φ f φ (t, Zt , Ut ) = f (t, Zt , Ut ) for all t ∈ [0, T ]. As a consequence of ¯ uniqueness of the solution of the JBSDE (2.18), we obtain that Y φ = Y . Now by Lemma 2.17  R ¯ c and part a) of Theorem 2.16 follows πtu (X) = Yt = Ytφ = ρt X − tT φ¯tr s dWs , t ∈ [0, T ]. To conclude that (2.23) holds, it remains to show that for any φ ∈ Φ one has Y ≤ Y φ . Let φ be in Φ. Then  f φ (Z φ , Ut ) − f φ (Z φ , U φ ) =(Z φ − φt )tr β˜t (Z φ − φ, U ) − β˜t (Z φ − φ, U φ ) t

t

t

t

t

t

Z

Ut (e)˜ γt (Z φ − φ, U )(e)λt (de)

+ ZE



E

(2.69)

Utφ (e)˜ γt (Z φ − φ, U φ )(e)λt (de).

 By part b) of Lemma 2.14, the couple γ˜t (Z φ − φ, U φ ), β˜t (Z φ − φ, U φ ) is the maximizer of R the expression (Ztφ − φt )tr βt + E Utφ (e)γt (e)λt (de) over all (γt , βt ) ∈ C¯t . Now by the fact  that γ˜t (Z φ − φ, U ), β˜t (Z φ − φ, U ) ∈ C¯t holds, it follows

(Ztφ



φt )tr β˜t (Z φ

φ

− φ, U ) +

Z E



(Ztφ



Utφ (e)˜ γt (Z φ − φ, U φ )(e)λt (de)

φt )tr β˜t (Z φ

− φ, U ) +

Z E

Utφ (e)˜ γt (Z φ − φ, U )(e)λt (de).

Using this inequality in (2.69) implies ftφ (Ztφ , Ut )



ftφ (Ztφ , Utφ )



Z E

γ˜t (Z φ − φ, U )(e)(Ut (e) − Utφ (e))λt (de).

(2.70)

By Assumption 2.13, (˜ γt (Z φ − φ, U ), β˜t (Z φ − φ, U )) is bounded in Rn × L2 (λt (ω)) uniformly in (t, ω). With this at hand, one shows using Proposition 2.31 and following the arguments in the  ˜ φ − φ, U ) · W + γ˜ (Z φ − φ, U ) ∗ µ e proof of Proposition 2.3 that the stochastic exponential E β(Z is a uniformly integrable martingale. Now applying Proposition 2.6 (plus Remark 2.7) to the JBSDEs with parameters (f1 , X1 ) := (f, X) and (f2 , X2 ) := (f φ , X) yields Yt ≤ Ytφ , t ∈ [0, T ]. To show the second claim of the theorem, let Qγ,β ∈ P ngd . The tracking error is given by ¯ ct . Using part a) of Theorem 2.16 and a change of measure Rφ (X) = πtu (X) − π0u (X) − φ¯ · W to Qγ,β , one obtains for all t ∈ [0, T ] that ¯ −dRtφ (X)

= f (t, Zt , Ut )dt − Zt dWt −

Z

ct ¯tr dW e(dt, de) + φ Ut (e)µ t

E



= f (t, Zt , Ut ) +

ξttr φ¯t

− (Zt − φ¯t )tr βt −

Z



Ut (e)γt (e)λt (de) dt E

Qγ,β

− (Zt − φ¯t )dWt



Z E

γ,β

eQ Ut (e)µ

(dt, de).

Section 2.5. Appendix

Page 94 ¯

Since by Lemma 2.18 we have f φ (t, Zt , Ut ) = ess infφ∈Φ f φ (t, Zt , Ut ) = f (t, Zt , Ut ), then ¯ the finite variation part under Qγ,β of Rφ (X) is non-decreasing and vanishes for (γ, β) = (γ ∗ , β ∗ ). By Assumption 2.13 k(γt (ω), βt (ω))kL2 (λt (ω))×Rn is bounded uniformly in (t, ω), ¯ e) ∈ S 2 . Since Rφ (X) ∈ S 2 , then H¨ hence dQγ,β /dP = E(β · W + γ ∗ µ older’s inequality implies ¯ ¯ Rφ (X) ∈ S 1 (Qγ,β ), and therefore Rφ (X) is a Qγ,β -supermartingale and a martingale under ∗ ∗ Q∗ = Qγ ,β . Proof of Lemma 2.22. One rewrites Cet (ω) = {(γ, β) ∈ Ht (ω) : G(γ, β) ∈ It (ω)} , where the map G : L2 (λ) × Rn → R is defined by G(γ, β) := kγk2L2 (λ) + |β|2 , and I and H are closed-convex-valued correspondences with values 

q



It (ω) = 0, Kt2 (ω) ⊆ R and Ht (ω) = γ ∈ L2 (λ) : γ ≥ − ζt (ω) × Rn ⊆ L2 (λ) × Rn . 



Since K is a predictable process, then I is predictable (and in particular P-measurable). Since G(·) is continuous, then applying two times [AF90, Theorem 8.2.9] (noting that L2 (λ) is a separable Hilbert space, hence a Polish space), one obtains first that H is P-measurable and then that Ce is P-measurable (since I is). Proof of Theorem 2.26. Without loss of generality, we argue for X ≥ 0, since otherwise one can use a translation argument with X + kXk∞ ≥ 0. Part 1: For t ∈ [0, T ], since Ctk (ω) ⊆ Ctk+1 (ω) ⊆ Ct (ω) for all k ∈ N, then πtu,k (X) ≤ πtu,k+1 (X) ≤ πtu (X), for any k ∈ N. Since X is bounded, then the monotone a.s. limit Jt := limk%∞ πtu,k (X) is finite and Jt ≤ πtu (X). It remains to show that πtu (X) ≤ Jt holds. To this end we show that J is a c`adl`ag Q-supermartingale for all Q ∈ Qngd and use Lemma 2.25. First J is a c`adl`ag Qγ,β -supermartingale for any Qγ,β ∈ Qngd with Girsanov kernel (γ, β) satisfying kγk2L2 (λt ) + |β|2 ≤ c, t ∈ [0, T ], for some constant c > 0. Indeed for such a measure Qγ,β , there exists k0 ∈ N such that (γ, β) ∈ C k for all k ≥ k0 . Since Jt = limk%∞,k≥k0 πtu,k (X) and π u,k (X) is a bounded c`adl`ag Qγ,β -supermartingale for every k ≥ k0 , then J is a c`adl`ag Qγ,β -supermartingale as the increasing limit of c`adl`ag Qγ,β -supermartingales of class D (see e.g. [Doo01, Section 2.IV.4]). This is in particular valid when Qγ,β ∈ Qngd for some k ∈ N. k γ,β ngd Now for an arbitrary Q ∈ Q , i.e. generally satisfying (γ, β = −ξ + η) ∈ C, define k k , t ∈ [0, T ]. Then the sequence (γ , η )k∈N with (γtk , ηtk ) := (γt , ηt )1k(γt ,ηt )k 2 n ≤k k k Qγ ,β

Qngd k .

L (λt )×R

= ∈ and hence := ∈ Moreover limk γ k = γ, P ⊗ λ ⊗ dta.e. and limk η k = η, P ⊗ dt-a.e. By the above argument, since ξ and X are bounded, then J b is a bounded c`adl`ag Q-supermartingale and hence admits a Doob-Meyer decomposition which, c, µ b F) (see e.g. [HWY92, e) under (Q, using weak predictable representation property of (W Theorem 3.22] or Part 2. in Example 1.1 of Chapter 1) and the fact that ν Qb = ν, reads c +U ∗µ b × H2 and A a non-decreasing predictable e − A, for (Z, U ) ∈ H2 (Q) J = J0 + Z · W ν

(γ k , β k

−ξ + η k )

Ck

Qk

Section 2.5. Appendix

Page 95

b processes with A0 = 0 and AT being Q-integrable since J is bounded (cf. [DM82, Inequality b to Qk on one hand and to Qγ,β on (15.1), Section VII.15]). By a change of measure from Q the other hand, one rewrites k

k

eQ + J = J0 + Z · W Q + U ∗ µ

Z · 0

Zttr ηtk +

Z



(2.71)

Ut (e)γtk (e)λt (de) dt − A,

E

and γ,β

J = J0 + Z · W Q

eQ +U ∗µ

γ,β

+

Z · 0

Zttr ηt +

Z



(2.72)

Ut (e)γt (e)λt (de) dt − A.

E

Now since (γ k , η k ) ∈ C k then J is a bounded c`adl`ag Qk -supermartingale for any k ∈ N. Hence  R tr k k from (2.71), it follows that dAt ≥ Zt ηt + E Ut (e)γt (e)λt (de) dt. Taking the limit as k







goes to ∞ and using the dominated convergence theorem (since η k ≤ |η| and γ k ≤ |γ|  R a.s.) one obtains dAt ≥ Zttr ηt + E Ut (e)γt (e)λt (de) dt, t ∈ [0, T ] and hence the process  R· R A − 0 Zttr ηt + E Ut (e)γt (e)λt (de) dt is non-decreasing and in particular non-negative. Now γ,β γ,β eQ since X is non-negative, so is J and this implies from (2.72) that J0 + Z · W Q + U ∗ µ is a non-negative local Qγ,β -martingale and is therefore a Qγ,β -supermartingale. Finally, Qγ,β  R R integrability of 0T Zttr ηt + E Ut (e)γt (e)λt (de) dt − AT follows, by boundedness of J, and thus J is a Qγ,β -supermartingale. Part 2: For k ∈ N, the process π u,k can be seen as the good-deal bound associated to the correspondence C k satisfying the hypotheses of Theorem 2.16, which implies the required result, after using the a-priori estimates of [Bec06, Proposition 3.3] to obtain Y ∈ S ∞ . b ∈ Qngd ⊂ Qngd . Hence by Lemma 2.25, π u (X) and Part 3: For any k ≥ kξk∞ holds Q k b π u,k (X) (for k ≥ kξk∞ ) are bounded c`adl`ag Q-supermartingales since X is bounded. Thus they admit Doob-Meyer decompositions (2.48) and (2.49) respectively. Now since the triple (π u,k (X), Z k , U k ) solves the JBSDE (2.47), one obtains that Ak satisfies (2.50). Moreover b × H2 (Q) b and AT , Ak ∈ L2 (Q) b follows from arguments in the proof of Part 1. (Z, U ) ∈ H2 (Q) ν T b Fu ) as k → ∞ for Part 4: From Part 3., that Aku converges to Au weakly in L2 (Ω, Q, all u ∈ [0, T ] follows Theorem VII.18 and subsequent remark]. This applies  from [DM82,  u,k since the sequence π (X) is uniformly bounded by kXk∞ (and hence uniformly k≥kξk∞

of class D), and therefore Part 1. and dominated convergence imply that πuu,k (X) converges b Fu ) as k → ∞ for all u ∈ [0, T ]. Furthermore the convergences of the to πuu (X) in L2 (Ω, Q, R R k cu + u e(ds, de) converges sequences (πuu,k (X))k∈N and (Aku )k∈N imply that Z k · W 0 E Us (e)µ RuR 2 cu + b e to Z · W (e) µ (ds, de) weakly in L (Ω, Q, F ) for all u ∈ [0, T ]. By the weak U u 0 E s c b e) under Q, strong orthogonality and isometry, the predictable representation property of (W , µ required result follows. Part 5: By Lemma 2.10, π u (X) is a Q-supermartingale with terminal value X, for all Q ∈ Qngd . ¯ is in the L1 -closure of Qngd , then π u (X) is also a Q-supermartingale ¯ Since by assumption Q

Section 2.5. Appendix

Page 96 ¯

with terminal value X. This together with π0u (X) = E Q [X] implies that π·u (X) has constant ¯ ¯ Q-expectation and is therefore a Q-martingale. Further, quasi-left-continuity of π u (X) is clear ¯ since ν  λ⊗dt implies ν Q  λ⊗dt by part a) of Lemma 2.1. Since X ∈ L∞ , then to show that b for any p ∈ [1, ∞), it suffices by dominated convergence π u,k (X) converges to π u (X) in S p (Q) u,k u to show that supt∈[0,T ] |πt (X)−πt (X)| converges to 0 in probability. By part 1. we know that πtu,k (X) % πtu (X) a.s. for all t ∈ [0, T ]. Moreover, denoting by p Y the predictable projection of an integrable process Y relative to the filtration F, it holds that p π u,k (X)t % p π u (X)t for any t ∈ [0, T ]. Recall that for every uniformly integrable martingale M holds p M = M− , and that for a predictable process K one has p K = K. Now because π u,k (X) is a bounded u,k b and Ak is continuous, then holds p π u,k (X)t = πt− (X). Moreover by Q-supermartingale u quasi-left-continuity of π (X), one has that A is a continuous process (by [HWY92, Theorem u (X). As a consequence one has π u,k (X) % π u (X) a.s. for 5.50]) and hence p π u (X)t = πt− t− t− all t ∈ [0, T ]. Now by the extended Dini’s Lemma in [DM82, Page 185] uniform convergence in time follows, i.e. supt∈[0,T ] |πtu,k (X) − πtu (X)| & 0. To prove the remaining claims, note that for all k ∈ N holds E Qb [AkT ] ≤ E Qb [|πTu,k (X) − π0u,k (X)|] ≤ 2kXk∞ , which implies that supk∈N E Qb [

RT 0

|dAkt |] < ∞. Finally by [BP90, Corollary 2], the required claims follow.

Proof of Proposition 2.29. Part 1: If Qγ,β ∈ Me , then by Lemma 2.8 it follows that β = −ξ+η with the required properties, P and moreover − log Γγ,β = −Mt + 12 hM c it − s≤t (log(1 + ∆Ms ) − ∆Ms ), holds for t ∈ [0, T ]. t P Define the process V by Vt := s≤t (log(1 + ∆Ms ) − ∆Ms ). Since E[− log Γγ,β T ] < ∞ then by Proposition 2.27 the process − log Γγ,β is a P -submartingale of class (D) with a Doob-Meyer R decomposition − log Γγ,β = N γ,β + Aγ,β . Hence M , hM c i = 0· |βs |2 ds and V are locally P -integrable with et + g(1 + γ) ∗ νt , −Vt = (− log(1 + γ) + γ) ∗ µt = g(1 + γ) ∗ µt = g(1 + γ) ∗ µ

where the third equality is obtained from [JS03, Proposition II.1.28] and g ≥ 0. Hence one has et + 12 hM c it + g(1 + γ) ∗ νt . Now because M = M c + M d − log Γγ,β = −Mt + g(1 + γ) ∗ µ t e, it follows that with M c = β · W and M d = γ ∗ µ et + − log Γt = −β · Wt − log(1 + γ) ∗ µ |

{z

:=Ntγ,β

}

Z t 1 |0

2

2

|βs | +

Z



g(1 + γs (e))λs (de) ds, E

{z

:=Aγ,β t

}

e is locally P -integrable thanks to the local P -integrability of V where the process log(1 + γ) ∗ µ e, and an application of [JS03, Proposition II.1.28]. and γ ∗ µ

Part 2: For Q ∈ nQngd ⊂ Me , Proposition 2.28 implies that o(κQ − υ)(B) ≤ 0, for any B ∈ P. R Choosing B = 12 |β|2 + E g(1 + γ· (e))ζ· (e)λ(de) > 21 K 2 , the Fubini’s theorem (see e.g. [Coh13, Proposition 5.2.1]) gives B ∈ P and hence (κQ − υ)(B) ≤ 0 holds, which implies by

Section 2.5. Appendix

Page 97

definition of κQ and υ that B is a P ⊗dt-nullset. Thus 12 |β|2 + E g(1+γ· (e))ζ· (e)λ(de) ≤ 12 K 2 holds, which is equivalent to (2.56), since by Part 1. one has Πt (βt ) = ξt and Π⊥ t (βt ) = ηt . R

e e-integrable function, and let β be a predictable Part 3: Let γ > −1 be a P-measurable and µ ⊥ process with Πt (βt ) = ξt and Πt (βt ) = ηt , t ∈ [0, T ], such that (γ, β) satisfies the inequality R R 2 1 1 2 1 2 2 |β| + E g(1 + γ· (e))ζ· (e)λ(de) ≤ 2 K . Then E g(1 + γ· (e))ζ· (e)λ(de) ≤ 2 K and 2 2 |β| ≤ K . By boundedness of K, the couple (γ, β) defines from Proposition 2.3 the Girsanov kernels of a measure Q ∼ P . That Q ∈ Me follows from Lemma 2.8. For such a measure Q the integrability condition on (γ, β) directly implies from the last claim of Proposition 2.27 that (2.54) is satisfied; hence Q ∈ Qngd .

Proof of Lemma 2.30. From the discussion preceding the statement of the lemma, one already  T T −1/2 has Cet ⊆ l∈N Cetl , t ∈ [0, T ]. Now let t ≤ T and (γ, β) ∈ l∈N Cetl , i.e. ζt 1{ζt >0} γ, β ∈  −1/2 1{ζ >0} γ, β ∈ C¯t . Define the sequence C¯tl , for all l ∈ N. It is to be shown that ζ t

−1/2

γ k , β k , k ∈ N such that ζt

t

−1/2

1{ζt >0} γ ∨ (−1 + 1/k) and β k := β, k ∈ −1/2 −1/2 √ N. One has for all k ∈ N that ζt 1{ζt >0} γ k ≤ ζt 1{ζt >0} γ (since γ ≥ − ζt ) and 

1{ζt >0} γ k := ζt −1/2

so dominated convergence implies that ζt



−1/2

1{ζt >0} γ k , β k converges to ζt 1{ζt >0} γ, β  −1/2 2 n in L (λt ) × R as k → ∞. To conclude that ζt 1{ζt >0} γ, β ∈ C¯t it remains to show −1/2

that ζt





1{ζt >0} γ k , β ∈ Ct holds for any k ∈ N. For this purpose, one has to show 

−1/2 −1/2 that 2g(1 + ζt 1{ζt >0} γ k ) L1 (λt ) + |β|2 ≤ Kt2 holds. Note that g 1 + ζt 1{ζt >0} γ k =



−1/2

g k 1+ζt

when 1 +

1{ζt >0} γ





−1/2

λ-a.e. since 1+ζt

−1/2 ζt 1{ζt >0} γ

−1/2

1{ζt >0} γ k either takes the value 1+ζt

1{ζt >0} γ −1/2 ≥ 1/k or 1/k when 1 + ζt 1{ζt >0} γ ≤ 1/k. This implies that

k

−1/2 −1/2 k

ζt 1{ζt >0} γ ) L1 (λt ) = 2g (1 + ζt 1{ζt >0} γ) L1 (λt ) . Since

for all k ∈ N, 2g(1 +  −1/2 ζt 1{ζt >0} γ, β ∈ C¯tk , for all k ∈ N, this concludes the first claim of the lemma.

To prove the second claim, recall that for any l ∈ N the function g l satisfies g l (1 + y) ≤ Const |y|2 for all y ≥ −1 for some Const > 0. We first show that the map Gl : [0, T ] ×

−1/2 Ω × L2 (λ) × Rn → R defined by Gl (t, ω, γ, β) := |β|2 + 2g l (1 + ζt (ω)γ) L1 (λt (ω))

is continuous in (γ, β) ∈ L2 (λ) × Rn . For this purpose, it suffices to show that for a R sequence (γ n )n ⊂ L2 (λ) which converges in L2 (λ) to γ, the sequence of integrals E g l (1 + R −1/2 n −1/2 ζt γ (e))λt (de) converges to E g l (1 + ζt γ(e))λt (de). If the measure λ is finite, this follows straightforwardly from the Lipschitz property of the function g l . For infinite λ, note that there exist by [AB06, Theorem 13.6] a subsequence (γ nk )k of (γ n )n and a function ψ ∈ L2 (λ) such that |γ nk | ≤ ψ λ-a.e. and γ nk converges λ-a.e. to γ. Renaming the R −1/2 nk γ (e))λt (de) converges to subsequence if necessary we can assume that E g l (1 + ζt R l R −1/2 n lim supn E g (1+ζt γ (e))λt (de) as k → ∞. Dominated convergence implies that E g l (1+ R R −1/2 nk −1/2 ζt γ (e))λt (de) converges to E g l (1 + ζt γ(e))λt (de), and therefore lim supn E g l (1 + R −1/2 n −1/2 ζt γ (e))λt (de) = E g l (1 + ζt γ(e))λt (de). By similar arguments one can also show

Section 2.5. Appendix

Page 98 −1/2

−1/2

that lim inf n E g l (1 + ζt γ n (e))λt (de) = E g l (1 + ζt γ(e))λt (de), and this concludes l the continuity of G . Now by Fubini’s theorem ([Coh13, Proposition 5.2.1]), the map Gl is e in addition P-measurable in (t, ω) ∈ [0, T ] × Ω since ζ is P-measurable and g l is continuous. Hence Gl is a Carath´eodory map and following the same arguments as those of the proof of Lemma 2.22 one shows that Ce l is P-measurable for any l ∈ N. Now P-measurability of Ce follows by [AF90, Theorem 8.2.4]. R

R

3. Hedging under generalized good-deal bounds and drift uncertainty In this chapter, we study good-deal valuation and hedging as in Chapter 2 but in a Brownian setting (i.e. continuous filtration), focusing on constructive examples with closed-form solutions and on robustness with respect to uncertainty (ambiguity) about the market price of risk (excess returns) of hedging assets. We describe robust good-deal bounds and hedging strategies by solutions to standard BSDEs, and show that robust good-deal hedging is equivalent to riskminimization (with respect to a specific no-good-deal pricing measure depending on the claim to be hedged) if uncertainty is very large. Section 3.1 formulates a framework for good-deal constraints which are described by predictable correspondences sufficiently general for all later sections to incorporate the natural radial (Sharpe ratio) constraints that are predominant in the good deal literature, but also extensions to ellipsoidal constraints. Section 3.2 studies hedging strategies and provides new examples with closed-form formulas for good-deal bounds and hedging strategies. In the presence of model uncertainty, good deal bounds and hedging strategies that are robust with respect to uncertainty are derived the Section 3.3, with the link to risk-minimization made more precise. In Appendix 3.4 we provide statements of intermediary results and proofs that are omitted in the main body of the chapter.

3.1

Mathematical framework and preliminaries

We work on a filtered probability space (Ω, F, F, P ) with time horizon T < ∞; the filtration F = (Ft )t≤T generated by an n-dimensional Brownian motion W , augmented with P -null-sets, satisfying the usual conditions. Let F = FT . Inequalities between random variables (processes) are meant to hold almost everywhere with respect to P (resp. P ⊗ dt). For stopping times τ ≤ T , the conditional expectation given Fτ under a probability measure Q is denoted by   EτQ · . We write Eτ = EτP if there is no ambiguity about P . Lp (Rm , Q), p ∈ [1, ∞), (or L∞ (Rm , Q)) denotes the space of FT -measurable Rm -valued random variables X with kXkpLp (Q) = E Q [|X|p ] < ∞ (resp. X Q-essentially bounded). P denotes the predictable σ-field on [0, T ] × Ω. Stochastic integrals of predictable integrands H with respect to semimartingales R p (Rm , Q) denote the space of predictable Rm S are denoted H · S = 0· Httr dSt . Let H h i valued processes Z with kZkpHp (Q) = E Q semimartingales Y with kY kS p (Q)

RT

|Zs |2 ds

0

= supt≤T |Yt |

p 2

< ∞, and S p (Q) that of c`adl`ag

< ∞ If the dimension is clear, we just

Lp (Q) Hp and

write and and if Q = P just S p , for p ∈ [1, ∞]. The Euclidean norm of a matrix M ∈ Rn×d is |M | := (Tr M M tr )1/2 and its usual operator norm is denoted by kM k. Lp (Q)

Hp (Q),

Lp ,

99

Section 3.1. Mathematical framework and preliminaries

Page 100

We will make use of classical theory of BSDEs [PP90, EPQ97]. BSDEs are stochastic differential equation of the type (3.1)

−dYt = f (t, Yt , Zt )dt − Zttr dWt , for t ≤ T, and YT = X,

where the terminal condition X is an FT -measurable random variable and the generator function f : Ω×[0, T ]×R1+n → R a P ⊗B(R1+n )-B(R)-measurable function. They are well established in mathematical economics. A pair (f, X) constitutes standard parameters (also called data) for a BSDE (3.1) if X ∈ L2 , f (·, 0, 0) is in H2 and f is uniformly Lipschitz in y and z, i.e. there exists L < ∞ such that |f (ω, t, y, z) − f (ω, t, y 0 , z 0 )| ≤ L(|y − y 0 | + |z − z 0 |) holds for all t, y, z, y 0 , z 0 . A solution of the BSDE (3.1) is a couple (Y, Z) of processes such that Y is R real-valued continuous, adapted, and Z is Rn -valued predictable and satisfies 0T |Zt |2 dt < ∞. For standard parameters (f, X) there exists a unique solution (Y, Z) ∈ S 2 × H2 to the BSDE (3.1), [EPQ97, Theorem 2.1]. Let us refer to BSDEs with standard parameters as classical and to the solution to such BSDEs as standard. A comparison theorem [EPQ97, Proposition 3.1] is very useful for optimal control problems stated in terms of classical BSDEs: Given standard BSDE solutions (Y, Z), (Y a , Z a )a∈A for a family of standard parameters (f, X), (f a , X a )a∈A , if there exists a ¯ ∈ A such that f (t, Yt , Zt ) = ess inf f a (t, Yt , Zt ) = f a¯ (t, Yt , Zt ), t ≤ T , and a∈A

X = ess inf X a = X a¯ , then Yt = ess inf Yta = Yta¯ holds for all t ≤ T. a∈A

a∈A

Section 3.1.1 will specify a financial market with d risky assets whose discounted price processes S i (i ≤ d) with respect to a fixed num´eraire asset (with unit price S 0 = 1) are non-negative locally bounded semimartingales. The set of equivalent local martingale measures (risk neutral pricing measures) is denoted by Me := Me (S) and we assume Me 6= ∅, i.e. there is no free lunch with vanishing risk in the sense of [DS94]. The market is incomplete with Me being of infinite cardinality if d < n. We will define generalized good-deal bounds by using abstract predictable correspondences C defined on [0, T ] × Ω with non-empty compact and convex values Ct (ω) ⊂ Rn , with predictability in the sense of [Roc76], i.e. for each closed set F ⊂ Rn , the set C −1 (F ) := {(t, ω) ∈ [0, T ] × Ω : Ct (ω) ∩ F 6= ∅} is predictable. More specific examples, e.g. for ellipsoidal constraints, will exhibit (semi)explicit solutions for optimizers. We write C : [0, T ] × Ω Rn with “ ” to emphasize that C is a set-valued mapping, and λ ∈ C to mean that the predictable function λ is a selection of C, i.e. λt (ω) ∈ Ct (ω) holds on [0, T ] × Ω. In the sequel a standard correspondence will refer to a predictable one, whose values are non-empty, compact and convex. Let C : [0, T ] × Ω Rn be a fixed standard correspondence with 0 ∈ C. The set Qngd := Qngd (S) of (equivalent) no-good-deal measures is given by n



o

Qngd (S) := Q ∈ Me dQ/dP = E (λ · W ) , λ predictable, bounded, λ ∈ C .

(3.2)

In the definition (3.2) and in subsequent definitions of sets of equivalent measures, we tacitly assume that Girsanov kernels λ are such that the stochastic exponentials E (λ · W ) are uniformly

Section 3.1. Mathematical framework and preliminaries

Page 101

integrable martingales. For good-deal valuation and hedging results later, concrete assumptions (e.g. Assumption 3.3) ensure that such holds for all selections λ of C. We remark that for good time-consistency properties, good-deal constraints should be specified locally in time ([KS07b]). For contingent claims X in L2 , upper and lower good-deal valuation bounds πtl (X) := ess inf EtQ [X] Q∈Qngd

and

πtu (X) := ess sup EtQ [X],

t ∈ [0, T ].

Q∈Qngd

(3.3)

are defined over a suitable (yet abstract) set of no good deal pricing measures Qngd . Hence πtu (X) (respectively πtl (X)) can be seen as the highest (lowest) valuation that does not permit too good deals to the seller (buyer). Since π·l (X) = −π·u (−X), further analysis can be restricted to π·u (X). As mentioned already in the introduction, the definition (3.3) in itself could already be viewed as a robust representation in a sense (over Q’s). For our purpose here however, the correspondence C and the respective set Qngd of no good deal measures are (at first) given with respect to one objective real world measure P (cf. remarks after (3.4)). To be clear in our use of terminology, we will in the sequel restrict our use of terms model uncertainty, ambiguity or robust hedging/valuation to situations with Knightian model uncertainty about P . Note that the use of terminology in some literature (e.g. [Del12]) is different, where the terms may instead refer to representations like (3.3). Definition (3.2) implies that density processes of measures Q ∈ Qngd are in S p , p ∈ [1, ∞). Hence X ∈ L2 = L2 (P ) ⊂ L1 (Q). In particular for X ∈ L∞ ⊂ L2 , we will show (cf. Theorem 3.7 and Proposition 3.5) that πtu (X) = ess supQ∈Qngd EtQ [X], where n



o

Qngd := Q ∈ Me dQ/dP = E (λ · W ) , λ predictable and λ ∈ C .

(3.4)

is a larger set than Qngd , containing measures with Girsanov kernels that are not necessarily bounded. We recall that for radial constraints C (like in (3.21) with A ≡ IdRn and constant h ∈ (0, ∞)), common in the good-deal literature, one has a known financial justification. By a direct duality argument, one can see (e.g. [Bec09, Section 3] in a semimartingale framework) that any (arbitrage-free) extension S¯ = (S, S 0 ) of the market S by derivative price processes S 0 := EtQ [X] for contingent claims X (with Q ∈ Me , X − ∈ L∞ , X + ∈ L1 (Q)) does permit ¯ whose expected only wealth processes V > 0 from self-financing trading strategy (in S) growth over h rates i (log utilities) h i any time period 0 ≤ t < τ ≤ T satisfy the (sharp) estimate EtP log VVτt ≤ EtP − log ZZτt , where Z is the density process of Q. For Q ∈ Qngd with radial constraint, this estimate is bounded by h2 (τ − t)/2, ensuring a bound h2 /2 to expected growth rates (good deals) for any market extension (ideas going back at least to [CR00, CH02]). For the good-deal bounds to have nice dynamic properties, multiplicative stability (m-stability) of the set of no-good-deal measures is important. M-stability of dominated families of probability measures in dual representations (like e.g. (3.3)) for dynamic coherent risk measures ( see e.g. [ADE+ 07]) ensures in particular time consistency (recursiveness) and has been studied

Section 3.1. Mathematical framework and preliminaries

Page 102

in a general context by [Del06]. In economics, it is known as rectangularity [CE02]. A set Q of measures Q ∼ P is called m-stable if for all Q1 , Q2 ∈ Q with density processes Z 1 , Z 2 and for all stopping times τ ≤ T , the process Z := I[0,τ ] Z·1 + I]τ,T ] Zτ1 Z·2 /Zτ2 is the density process of a measure in Q, where [0, τ ] := {(t, ω) ∈ [0, T ] × Ω | t ≤ τ (ω)} denotes the stochastic interval and IA is the indicator function on a set A. As noted in [Del06, Rem. 6], by closure this definition extends to sets of measures that are absolutely continuous but not necessarily equivalent; such is formally achieved by setting ZT2 /Zτ2 = 1 on {Zτ2 = 0}. The role of m-stability shows in results due to [Del06], stated in Lemma 3.1, Part a); for details cf. [KS07b, Theorem 2.7] or [Bec09, Proposition 2.6]. Proof for part b) is provided in the appendix. Lemma 3.1. Let Q be a convex and m-stable set of probability measures Q ∼ P and πtu,Q (X) := ess supQ∈Q EtQ [X], for X ∈ L∞ . a) There exists a c`adl`ag version Y of π·u,Q (X) such that for all stopping times τ ≤ T , Yτ = ess supQ∈Q EτQ [X] =: πτu,Q (X). Moreover π·u,Q (·) has the properties of a dynamic coherent risk measure. It is recursive and stopping time consistent: For stopping times   u,Q 1 u,Q u,Q 1 u,Q σ ≤ τ ≤ T holds πσ (X ) = πσ πτ (X ) , and πτ (X 1 ) ≥ πτu,Q (X 2 ) for X 1 , X 2 ∈ L∞ implies πσu,Q (X 1 ) ≥ πσu,Q (X 2 ). Finally, a supermartingale property holds: For all stopping   times σ ≤ τ ≤ T and Q ∈ Q, πσu,Q (X) ≥ EσQ πτu,Q (X) , and π·u,Q (X) is a supermartingale under any Q ∈ Q. b) The sets Me and Qngd are m-stable and convex and hence for Q = Qngd , π·u (X) = π·u,Q (X) satisfies the properties of Part a).

3.1.1

Parametrizations in an Itˆ o process model

This section describes the Itˆo process framework for the financial market, and details the parametrizations for dynamic trading strategies and for the no-good-deal constraints. The latter are specified at this stage by abstract correspondences (3.2) such that respective dynamic no-good-deal valuation bounds for contingent claims can be conveniently described in terms of (super-)solutions to BSDEs (Sections 3.1.2-3.1.3) within a convenient framework sufficiently general for all later Sections 3.2-3.3. We consider models for financial markets where prices (S i )i=1...d of d risky assets evolve according to a stochastic differential equation (SDE) ct , t ∈ [0, T ], dSt = diag(St )σt (ξt dt + dWt ) =: diag(St )σt dW

S0 ∈ (0, ∞)d ,

for predictable Rd - and Rd×n -valued coefficients ξ and σ, with d ≤ n. This includes basically all examples of continuous price and state evolutions in (typically incomplete) markets of the good-deal literature, and permits also for non-Markovian evolutions. Risky asset prices S are

Section 3.1. Mathematical framework and preliminaries

Page 103

given in units of some riskless num´eraire asset whose discounted price S 0 ≡ 1 is constant. We assume that σ is of maximal rank d ≤ n (i.e. det(σt σttr 6= 0, that means no locally redundant assets) and that the market price of risk process ξ, satisfying ξt ∈ Im σttr , is bounded. This ensures that market is free of arbitrage but typically incomplete (if d < n) in the sense that b given by dQ b = E (−ξ · W ) dP (see Me 6= ∅, as the minimal local martingale measure Q [Sch01]) is in Me , which however is typically not a singleton. Trading strategies are represented by the amount of wealth ϕ = (ϕit )i invested in the risky assets (S i )i . A self-financing trading strategy is described by a pair (V0 , ϕ), where V0 is the initial capital while ϕ = (ϕit )i describes the amount of wealth invested in the risky assets (S i )i at any time t. Theh set Φϕ of permitted i R 2 strategies consists of Rd -valued predictable processes ϕ satisfying E P 0T |ϕtr t σt | dt < ∞. For an permitted strategy ϕ, the associated wealth process V from initial capital V0 has c dynamics dVt = ϕtr t σt dWt . To ease notation, we re-parametrize strategies in Φϕ in terms of c . Indeed, equalities φ = σ tr ϕ and ϕ = (σ tr )−1 φ, where integrands φ := σ tr ϕ with respect to W (σ tr )−1 := (σσ tr )−1 σ is the pseudo-inverse of σ tr , provide a one-to-one relation between ϕ and φ. Define the correspondences Γt (ω) := Im σttr (ω) and Γ⊥ t (ω) := Ker σt (ω),

(t, ω) ∈ [0, T ] × Ω,

(3.5)

where Im σttr and Ker σt denote the range (image) and the kernel of the respective matrices. n Clearly, Rn = Γt ⊕ Γ⊥ t and any z ∈ R decomposes uniquely into its orthogonal projections as ⊥ z = ΠΓt (z) ⊕ ΠΓ⊥ (z) =: Πt (z) ⊕ Πt (z). Let t

(

Φ = Φφ :=

hZ φ φ is predictable, φ ∈ Γ and E

T

i

)

|φt |2 dt < ∞

0

denote the (re-parametrized) set of permitted trading strategies. Proving the claims of the next proposition is routine, using [Roc76] for the first. Proposition 3.2. dictable.

1. The correspondences Γ and Γ⊥ are closed-convex-valued and pre-

2. Q ∈ Me if and only if Q ∼ P with dQ = E(λ · W )dP , where λ is predictable and λ = −ξ + η, with −ξt = Πt (λt ) ∈ Im σttr and ηt = Π⊥ t (λt ) ∈ Ker σt ∀t. By Part 2 of Proposition 3.2, the set Qngd defined in (3.2) can be written as n



o

Qngd = Q ∼ P dQ/dP = E (λ · W ) , λ predictable, bounded and λ ∈ Λ ,

(3.6)

where Λ : [0, T ] × Ω Rn is defined by Λt (ω) := Ct (ω) ∩ (−ξt (ω) + Ker σ). By Part 1 of Proposition 3.2 and [Roc76, Corollary 1.K and Theorem 1.M], Λ is a compact-convex-valued predictable correspondence. Slightly beyond the no-free-lunch with vanishing risk condition, b or equivalently −ξ ∈ C. This implies that Λ is we assume that Qngd contains the measure Q, non-empty valued, hence standard.

Section 3.1. Mathematical framework and preliminaries

3.1.2

Page 104

Good-deal valuation with uniformly bounded correspondences

We here consider the case where the no-good-deal constraint is described by a uniformly bounded correspondence; a more general case is studied afterwards. We say that a correspondence C is uniformly bounded if it satisfies Assumption 3.3. sup(t,ω) supx∈Ct (ω) |x| < ∞. Let C : [0, T ] × Ω Rn be a standard correspondence satisfying Assumption 3.3 and 0 ∈ C. Under Assumption 3.3, selections of C are uniformly bounded processes. In particular, the Girsanov kernels of no-good-deal measures are uniformly bounded, and hence boundedness in the definition (3.2) (see also (3.6)) of Qngd is not necessary. The good-deal valuation bound πtu (X) := ess supQ∈Qngd EtQ [X] is well-defined for a contingent claim X ∈ L2 ⊃ L∞ , that may be path-dependent, and one can check that in this case an analog of Part a) of Lemma 3.1 still holds. Though Assumption 3.3 fits well with the classical theory, it would be too restrictive to impose it in general since it may not hold in some interesting practical situations; see for instance the example in Section 3.2.2. Let us recall a fact about linear BSDEs (cf. [EPQ97]) which explains their role for valuation purposes. Lemma 3.4. For Q ∼ P with bounded Girsanov kernel λ, the linear BSDE −dYt = Zttr λt dt − Zttr dWt , t ≤ T,

with YT = X in L2 ,

(3.7)

has a unique standard solution (Y λ , Z λ ) with Ytλ = EtQ [X] = Y0λ + Z · WtQ , and W Q := R W − 0· λt dt. If X ∈ L∞ then Y is bounded. Boundedness of λ in Lemma 3.4 clearly implies that the parameters of the BSDE (3.7) are standard. For unbounded λ, the classical BSDE theory no longer applies and one needs different results to characterize the good-deal bounds in terms of BSDEs. Under Assumption 3.3, Λ is uniformly bounded and thus Girsanov kernels λQ for all Q ∈ Qngd are bounded by the same constant. One has the following Proposition 3.5. Let Assumption 3.3 hold. ¯ := λ(Z) ¯ 1. For any predictable Rn -valued process Z, there exists a predictable process λ = tr tr ¯ ¯ (λt (Zt ))t≤T ∈ Λ such that λt Zt = ess sup λt Zt , t ∈ [0, T ]. λt ∈Λt

2. For X ∈ L2 , let (Y λ , Z λ ) (for λ = λQ ∈ Λ, Q ∈ Qngd ) and (Y, Z) be respectively standard solutions to the classical BSDEs (3.7) and ¯ t (Zt )dt − Z tr dWt , t ≤ T, and −dYt = Zttr λ t

YT = X,

(3.8)

¯ ¯ = λ(Z) ¯ with λ from Part 1. Then πtu (X) = ess supQ∈Qngd EtQ [X] = EtQ [X] = Yt holds ¯ λ λ ¯ · W )dP , Yt = ess sup ¯ ∈ Qngd given by dQ ¯ = E(λ for Q λ∈Λ Yt = Yt .

Section 3.1. Mathematical framework and preliminaries

Page 105

Proof. Part 1 follows by a direct application of the measurable maximum theorem [Roc76, Theorem 2.K] and measurable selection theorem [Roc76, Theorem 1.C]. As for Part 2, by ¯ Assumption 3.3 the parameters of the BSDEs (3.8) and (3.7) are standard. Moreover Q ngd ¯ ∈ Λ. The remaining of Part 2 hence follows from existence and is clearly in Q since λ uniqueness results as well as the comparison theorem for classical BSDEs, cf. [EPQ97, Section 2-3]

3.1.3

Good-deal valuation with non-uniformly bounded correspondences

To relax the Assumption 3.3 of uniform boundedness, we now admit for a non-uniformly bounded standard correspondence C, with 0 ∈ C, which satisfies ∃ R predictable with

sup |x| ≤ Rt (ω) ∀(t, ω) and x∈Ct (ω)

Z

T

|Rt |2 dt < ∞.

0

(3.9)

It is relevant to look beyond Assumption 3.3, because examples of practical interest require to do so, see Section 3.2.2 where quasi-explicit formulas of good-deal bounds are obtained in a stochastic volatility model, with C not being uniformly bounded but satisfying (3.9). Classical BSDE results do not apply as before to characterize good-deal bounds directly by standard BSDE solutions. Yet, we can still (cf. Theorem 3.7) approximate π·u (X) for X ∈ L∞ by solutions to classical BSDEs for suitable truncations of C, and prove that πtu (X) coincides with the essential supremum over the larger set Qngd ⊆ Me given in (3.4). We show, under condition (3.9), that π·u (X) is the minimal supersolution of the BSDE (3.8). Finally, we ¯ for π u (X) exists. show that π·u (X) is the minimal solution to (3.8) if a worst-case measure Q 0 ¯ may be attained rather in the larger set Qngd . Obviously, a maximizing Q To this end, let Ctk (ω) = {x ∈ Ct (ω) : |x| ≤ k} for (t, ω) ∈ [0, T ] × Ω with k ∈ N be a sequence of correspondences. Since C is standard with 0 ∈ C, the same holds for each C k . Clearly, any C k satisfies Assumption 3.3 and Ctk (ω) % Ct (ω) as k % ∞. For each k ∈ N, let Qngd := Qngd k k (S) denote the set n



Qngd := Q ∼ P dQ/dP = E (λ · W ) , with λ predictable and λ ∈ Λk k

o

(3.10)

of no-good-deal measures (for S) corresponding to C k with Λk : [0, T ] × Ω Rn given by Λkt (ω) := Ctk (ω) ∩ (−ξt (ω) + Γ⊥ t (ω)) and hence also satisfying Assumption 3.3. For 2 X ∈ L , we define analogously the bounds π·u,k (X) associated to the sets Qngd k , k ∈ N as u,k Q ngd πt (X) := ess supQ∈Qngd Et [X], t ∈ [0, T ]. The sets Qk , k ∈ N are m-stable and convex k as well.

Section 3.1. Mathematical framework and preliminaries

Page 106

Lemma 3.6. (Dynamic principle): Let Q be a convex and m-stable set of probability measures Q ∼ P and πtu,Q (X) := ess supQ∈Q EtQ [X], for X ∈ L∞ . Then π·u,Q (X) is the smallest adapted c`adl`ag process that is a supermartingale under any Q ∈ Q with terminal value X. Proof. The supermartingale properties of π·u,Q (X) under every Q ∈ Q hold by Part a) of Lemma 3.1. Let Y be another process satisfying the same properties. Then for all Q ∈ Q one has Yt ≥ EtQ [X], t ∈ [0, T ], and taking the essential supremum over Q ∈ Q then yields Yt ≥ πtu,Q (X).

ngd are convex and m-stable, Lemma 3.6 holds in particular for Note that since Qngd k , Q u,Qngd

ngd

π·u (X) = π·u,Q (X) and π·u,k (X) = π· k (X), k ∈ N. Theorem 3.7 is analogous to Parts 1.-4. of Theorem 2.26 in Chapter 2. We still include its proof in Appendix 3.4, because it seems more instructive in the absence of jumps. Theorem 3.7. For any contingent claim X ∈ L∞ it holds 1. πtu,k (X) % ess sup EtQ [X] = πtu (X) P -a.s. as k % ∞, for all t ∈ [0, T ]. 2. For any k

Q∈Qngd ∈ N, π·u,k (X)

= Y k for (Y k , Z k ) being standard solution to the BSDE

tr −dYt = (ess sup λtr t Zt )dt − Zt dWt , t ≤ T, with

YT = X.

λt ∈Λkt

(3.11)

3. π·u (X) and π·u,k (X) for k ≥ kξk∞ admit Doob-Meyer decompositions c −A π·u (X) = π0u (X) + Z · W

and

c − Ak , π·u,k (X) = π0u,k (X) + Z k · W

(3.12)

b , where Z, Z k ∈ H2 (Q) b and A, Ak are non-decreasing predictable processes with under Q k 2 k b AT , AT ∈ L (Q), A0 = A0 = 0 and

Ak =

Z · 0



k ξttr Ztk + ess sup λtr t Zt dt. λt ∈Λkt

(3.13)

b Fu ) and Z k → Z weakly in L2 (Ω×[0, u], Q⊗dt). b 4. For all u ≤ T , Aku → Au weakly in L2 (Ω, Q,

Let g be the function defined by gt (z) := ess sup λtr t z, λt ∈Λt

t ∈ [0, T ], z ∈ Rn .

(3.14)

Since g may not be Lipschitz if C does not satisfy Assumption 3.3, then π·u (X) cannot directly be characterized by classical BSDEs. But one can still obtain a characterization by the minimal supersolution to the BSDE with data (g, X).

Section 3.1. Mathematical framework and preliminaries

Page 107

Definition 3.8. (Y, Z, K) is a supersolution of the BSDE with parameters (f, X) if −dYt = f (t, Yt , Zt )dt − Zttr dWt + dKt

for t ≤ T, and YT = X,

with K non-decreasing c`adl`ag adapted, , K0 = 0, and 0T |Zt |2 dt < ∞. A supersolution with K ≡ 0 is a BSDE solution. A (super)solution (Y, Z, K) is minimal if Yt ≤ Y¯t , t ∈ [0, T ] holds ¯ K). ¯ for any other (super)solution. (Y¯ , Z, R

Note that a minimal supersolution when it exists is unique, as minimality implies uniqueness of the Y -components; since continuous local martingales of finite variation are trivial, identity of the Z- and K-components follows. Existence of the minimal supersolution is sometimes investigated under the condition that there exists at least one supersolution to the BSDE (cf. [DHK13]). This condition is satisfied for the BSDE with parameters (g, X), X ∈ L∞ since g(·, 0) = 0 and thus (Y, Z, K) := (|X|∞ − (|X|∞ − X)I{T } , 0, (|X|∞ − X)I{T } ) is a supersolution. Note that g satisfies gt (z) ≥ −ξttr z, t ∈ [0, T ] and moreover (g, X) satisfies the hypotheses of [DHK13, Theorem 4.17] which implies existence of the minimal supersolution to the BSDE with parameter (g, X). We show that π·u (X) can be identified with the Y -component R of this minimal supersolution. Condition (3.9) ensures that the process 0· gt (Zt )dt for g in R (3.14) and Z satisfying 0T |Zt |2 dt < ∞ is real-valued, since Cauchy-Schwarz inequality would R R 1 1 R imply 0T |gt (Zt )|dt ≤ ( 0T |Zt |2 dt) 2 ( 0T |Rt |2 dt) 2 < ∞. b and a non-decreasing Theorem 3.9. Let (3.9) hold and X ∈ L∞ . There exists Z ∈ H2 (Q) u predictable process K with K0 = 0 such that (π· (X), Z, K) is the minimal supersolution to the BSDE for data (g, X) with g from (3.14), and π·u (X) ∈ S ∞ .

The proofs for this theorem and for the next corollary are given in Appendix 3.4. ¯ ∈ Qngd such Corollary 3.10. Let (3.9) hold and X ∈ L∞ . If there exists a measure Q ¯ u Q Q u ¯ that π0 (X) = supQ∈Qngd E [X] = E [X], then π· (X) is a Q-martingale and there exists 2 u b Z ∈ H (Q) such that (π· (X), Z) is the minimal solution to the BSDE with parameters (g, X) ¯ of Q ¯ tr Zt , for all ¯ satisfies ess supλ ∈Λ λtr Zt = λ for g defined in (3.14). The Girsanov kernel λ t t t t t ∈ [0, T ]. ¯ ∈ Qngd as in Corollary 3.10 may be shown by For concrete case studies, existence of Q direct considerations, see Section 3.2.2 for examples. If one could formulate the no-good¯ would exist for deal restriction so that the set Qngd becomes weakly compact in L1 , then Q ∞ any X ∈ L from maximizing a bounded linear objective functional over a weakly compact subset of L1 . Note that Assumption 3.3 only implies (by Dunford-Pettis compactness theorem [DM78, Chapter II, Theorem 25]) that Qngd is weakly relatively compact in L1 . If Qngd is not weakly relatively compact in L1 , then by James’ theorem (cf. [AB06, Theorem 6.36]) there exists X ∈ L∞ such that the supremum in π0u (X) = supQ∈Qngd E Q [X] is not attained in the

Section 3.2. Dynamic good-deal hedging

Page 108

L1 -closure of Qngd (since Qngd is convex) and in particular also not in Qngd . Let us give ¯ does not exist in Qngd for some contingent claim and C does neither an example where Q satisfy Assumption 3.3 nor (3.9). Section 3.2.2 will furthermore give an example in a stochastic ¯ exists and C is not uniformly bounded but satisfies (3.9). volatility model where Q Example 3.11. Let n = 2 with W = (W 1 , W 2 ), d = 1 with dSt = St σ S dWt1 , S0 > 0, R σ S > 0, and ξ = 0. Let h > 0 be a deterministic predictable process with 0T ht dt = ∞ and Ct (ω) := {0} × [−ht , ht ], (t, ω) ∈ [0, T ] × Ω. Now let X := I{W 2 ≥0} ∈ L∞ , then T π0 := supn∈N Qn [{WT2 ≥ 0}] ≤ π0u (X) ≤ 1, where dQn = E(λn · W 2 )dP with λnt = R ht ∧ n, t ∈ [0, T ], n ∈ N. The process W 2,n := W 2 − 0· λnt dt is a Qn -Brownian motion. R R Hence WT2,n ∼ N (0, T ) under Qn . We have 0T λnt dt % 0T ht dt = ∞ as n % ∞. Hence R π0 = supn∈N Qn [{WT2,n ≥ − 0T λnt dt}] = 1. Therefore π0u (X) = 1. But there exists no ¯ ∈ Qngd such that π u (X) = E Q¯ [X]. Indeed for such a measure, one would have measure Q 0 0 2 ≥ 0}] = 1 which is not possible since Q ¯ ¯ ∼ P. Q[{W T

3.2

Dynamic good-deal hedging

Let again C be a standard correspondence satisfying 0 ∈ C, and define the family of a-priori valuation measures n



P ngd := Q ∼ P dQ/dP = E (λ · W ) , λ predictable, bounded, λ ∈ C

o

(3.15)

which satisfy the same no-good-deal constraint as those in Qngd , except that the local martingale condition for S is omitted. One could view P ngd as the no-good-deal measures for a market only consisting of the riskless asset S 0 ≡ 1, i.e. P ngd = Qngd (1). It is natural to define (3.15) as a-priori valuation measures, as the concept of no-good-deal valuation is to consider those risk neutral valuation measures Q, for which any extension of the financial market by additional derivatives price processes (being Q-martingales) would not give rise to ’good deals’; see e.g. [BS06, KS07a, Bec09] for rigorous detail in continuous time for Sharpe ratios, utilities or growth rates; for concepts cf. [Cer03]. Like Qngd , the set P ngd clearly is again m-stable and convex. Just as in (3.3), we define the a-priori dynamic coherent risk measure ρt (X) := ess sup EtQ [X], Q∈P ngd

t ∈ [0, T ],

(3.16)

for contingent claims X ∈ L2 . Note that ρt (X) is well-defined as the essential supremum of finitely valued random variables since measures in P ngd have bounded Girsanov kernels and hence density processes in S p (P ) for any p ∈ [1, ∞). Elements Q of P ngd or Qngd can be considered as generalized scenarios (as in [ADE+ 07]). Since P ngd ∩ Me = Qngd clearly holds, then ρt (X) ≥ πtu (X) for all t ≤ T . An investor holding

Section 3.2. Dynamic good-deal hedging

Page 109

a liability X and trading in the market according to a permitted trading strategy φ, would R c assign at time t a residual risk ρt (X − tT φtr s dWs ) to his position. The investor’s objective is to hedge his position by a trading strategy φ¯ that minimizes his residual risk at any time t ≤ T . To justify a premium π·u (X) for selling X, the minimal capital requirement to make his position ρ-acceptable should coincide with π·u (X). Thus, his hedging problem is to find a trading strategy φ¯ ∈ Φ such that 

πtu (X) = ρt X −

Z t

T





c φ¯tr s dWs = ess inf ρt X − φ∈Φ

Z t

T



c φtr s dWs ,

(3.17)

t ∈ [0, T ].

The good-deal hedging strategy will be defined as a minimizer φ¯ in (3.17), and the good-deal valuation π·u (·) becomes the market consistent risk measure corresponding to ρ, in the spirit of [BE09]. For a contingent claim X, the tracking error at time t ∈ [0, T ] ct Rtφ (X) := πtu (X) − π0u (X) − φ · W

(3.18)

of a hedging strategy φ ∈ Φ is defined as the difference between the dynamic variations in the capital requirement and the profit/loss from trading (hedging) according to φ up to time t. Proposition 3.12. For X ∈ L2 , let the strategy φ¯ ∈ Φ solve (3.17). Then the tracking error ¯ Rφ (X) is a Q-supermartingale for all Q ∈ P ngd . ¯

Proof. By the first equality of (3.17) and the definition of the tracking error it holds Rtφ (X) = R c −π0u (X) + ess supQ∈P ngd EtQ [X − 0T φ¯tr s dWs ], t ∈ [0, T ]. The claim then follows from m-stability and convexity of P ngd , applying Lemma 3.1, Part a) extended to X ∈ L2 by some BSDEs arguments. From Remark 2.12 in Chapter 2, let us point out that by Proposition 3.12 we can view the good-deal hedging strategy as being at least mean-self-financing under Q ∈ P ngd . The latter is a property that we again interpret as robustness of φ¯ with respect to the set of measure P ngd as generalized scenario (in the sense of [ADE+ 07]). To describe solutions to the hedging problem (3.17), we will often assume that C has further structure and is uniformly bounded. Section 3.2.2 also contains an example for a correspondence C that is not uniformly bounded but satisfies (3.9) in the Heston model, where the hedging problem can be solved in a semi-explicit manner. For a correspondence C satisfying Assumption 3.3, one can describe ρ· (X) (like π·u (X) in Proposition 3.5, proof being analogous) by solutions to classical BSDEs: e and (Y λ , Z λ ) (for λ ∈ C) Proposition 3.13. Let Assumption 3.3 hold. For X ∈ L2 , let (Ye , Z) be the respective standard solutions to the BSDEs

˜ t dt − Z tr dWt , t ≤ T, with YT = X, and −dYt = Zttr λ t

(3.19)

−dYt = Zttr λt dt − Zttr dWt , t ≤ T, with YT = X,

(3.20)

Section 3.2. Dynamic good-deal hedging

Page 110

tr ˜ = λ(Z) ˜ ˜ tr Zt = ess sup where λ ∈ C is a predictable process satisfying the equality λ λt ∈Ct λt Zt t e e with Girsanov kernel λQ ˜ is in P ngd , and ρt (X) = for t ∈ [0, T ]. Then the measure Q =λ

e ess sup Ytλ = EtQ [X] = Y˜t , t ∈ [0, T ]. λ∈C

3.2.1

Results for ellipsoidal no-good-deal constraints

This section derives more explicit BSDE results to describe the solution to the valuation and the hedging problem (3.17) for (predictable) ellipsoidal no-good-deal constraints. Such generalization includes the important special case of radial constraints (as e.g. in [Bec09]), which is common to the good-deal literature and justified by bounds (uniform in (t, ω)) on optimal growth rates or instantaneous Sharpe ratios, while still permitting comparably explicit results. The generalization could be interpreted as imposing different bounds on growth rates (or Sharpe ratios) for the risk factors associated to the principal axes. While such might appear as rather technical at this stage, in the subsequent context of model uncertainty (cf. Remark 3.24 b)) non-radial constraints will appear naturally. To this end, let h be a positive bounded predictable process, and A be a predictable Rn×n -matrixvalued process with symmetric values and uniformly elliptic i.e. Atr = A and xtr Ax ≥ c |x|2 , for all x ∈ Rn and some c ∈ R+ . The common radial case is achieved by choosing A ≡ IdRn . We define the standard (see [Roc76, Corollary 1.Q]) correspondence n

o

Ct (ω) = x ∈ Rn | xtr At (ω)x ≤ h2t (ω) ,

(t, ω) ∈ [0, T ] × Ω,

(3.21)

that satisfies Assumption 3.3 due to ellipticity and boundedness of h. Assume that the kernel of the volatility matrix σ is spanned by eigenvectors of A, i.e. A−1 t (Ker σt ) = Ker σt ,

(3.22)

t ∈ [0, T ].

As the eigenvectors of A are orthogonal and (Ker σ)⊥ = Im σ tr , then (3.22) can be interpreted as separability of Im σ tr and Ker σ in the sense that each of these subspaces has a basis of eigenvectors of A. Given (3.22), the subspaces Im σ tr and Ker σ are orthogonal under the scalar product defined by A, one can re-write

n

o

Qngd = Q ∼ P dQ/dP = E (λ · W ) , λ predictable, λ = −ξ + η, η ∈ C ξ ∩ Ker σ , with Ctξ (ω) = x ∈ Rn | xtr At (ω)x ≤ h2t (ω) − ξt (ω)tr At (ω)ξt (ω) , also satisfying Assumption 3.3. The correspondence C ξ is standard if 



h2 > ξ tr Aξ,

(3.23)

The separability condition (3.22) ensures that −ξ + η ∈ C is equivalent to η ∈ C ξ , for η ∈ Ker σ. This way the ellipsoidal constraint on the Girsanov kernels transfers to one on their

Section 3.2. Dynamic good-deal hedging

Page 111

η-component, which permits to formulate the no-good-deal constraint only with respect to non-traded risk factors in the market. In this setup, it is straightforward to obtain an expression ¯ from Part 1 of Proposition 3.5 via for λ Lemma 3.14. For z ∈ Rn \ {0}, h > 0 and a symmetric positive definite n × n-matrix A, the unique maximizer of y tr z subject to y tr Ay ≤ h2 is y¯ = h(z tr A−1 z)−1/2 A−1 z. For X ∈ L2 , since C satisfies Assumption 3.3, there exists a unique standard solution (Y, Z) to the BSDE with terminal condition YT = X and 

dYt =

ξttr Πt (Zt )

q



h2t

ξttr At ξt





q

tr −1 ⊥ Π⊥ t (Zt ) At Πt (Zt )

dt + Zttr dWt .

(3.24)

¯ from Part 1 of We will see that π·u (X) = Y holds, and that the optimal Girsanov kernel λ ¯ Proposition 3.5 takes the form λ = −ξ + η¯ with η¯ ∈ Ker σ given by q

h2t − ξttr At ξt

η¯t = q

tr

Π⊥ t (Zt )

⊥ A−1 t Πt (Zt )

⊥ A−1 t Πt (Zt ),

t ∈ [0, T ].

(3.25)

In particular when h tends to ξ tr Aξ P ⊗ dt-a.s., then η¯ tends to 0 and the good-deal bound b is the minimal local martingale measure. By Lemma πtu (X) converges to EtQ [X], for Q tr ⊥ 3.14 and using (3.22), one obtains η¯ttr Π⊥ t (Zt ) = ess supηt ∈C ξ ∩Ker σt ηt Πt (Zt ), and hence b

t

¯ tr Zt = −ξ tr Πt (Zt ) + h2 − ξ tr At ξt 1/2 Π⊥ (Zt )tr A−1 Π⊥ (Zt )1/2 , t ≤ T . Therefore Part 2 λ t t t t t t t of Proposition 3.5 yields Theorem 3.15. Assume (3.22) and (3.23) hold. For X ∈ L2 , let (Y, Z) be the standard ¯ ¯ ∈ Qngd is solution to the BSDE (3.24). Then πtu (X) = Yt = EtQ [X], t ∈ [0, T ], where Q ¯ = E ((−ξ + η¯) · W ) dP with η¯ given explicitly by (3.25). given by dQ The observation of the following lemma is straightforward. Lemma 3.16. The matrices A−1 t (ω), for (t, ω) ∈ [0, T ] × Ω, are positive-definite and satisfy 0 (ω) |x|2 for all x, t, where α0 (ω) = ckA (ω)k−2 > 0 for c being the constant xtr A−1 (ω)x ≥ α t t t t of uniform ellipticity of A. Moreover kAk ≥ c holds. tr ˜ = h(Z tr A−1 Z)−1/2 A−1 Z satisfies λ ˜ tr Zt = ess sup tr By Lemma 3.14, λ t λt At λt ≤h2t λt Zt , ˜ tr Zt = ht (Z tr A−1 Zt )1/2 , t ∈ [0, T ]. Hence Proposition 3.13 gives ρt (X) = t ∈ [0, T ], with λ t t t Yt , t ∈ [0, T ], where (Y, Z) uniquely solves the classical BSDE with terminal condition YT = X and 1/2 −dYt = ht (Zttr A−1 dt − Zttr dWt . (3.26) t Zt )

Section 3.2. Dynamic good-deal hedging

Page 112

Thanks to Lemma 3.16, a sufficient condition to ensure (3.23) is √ |ξ| < h α0 .

(3.27)

In addition it is used to verify for Lemma 3.35 the Kuhn-Tucker conditions before applying the Kuhn-Tucker theorem (see [Roc70, Section 28]), after which comparison results for BSDE yield the result of Theorem 3.17 below. The proof is omitted as it is analogous to that of [Bec09, Theorem 5.4 and Lemma 6.1], using now Lemma 3.35 instead of Lemma 6.1 there. For φ ∈ Φ, let (Y φ , Z φ ) denote the standard solution to the BSDE with terminal condition YT = X and, for t ≤ T , −dYt =



− ξttr φt + ht (Zt − φt )tr A−1 t (Zt − φt )

1/2 

dt − Zttr dWt .

(3.28)

Theorem 3.17. Assume (3.22),(3.27) hold. For X ∈ L2 , let (Y, Z) and (Y φ , Z φ ) (for φ ∈ Φ) R c be standard solutions to the BSDEs (3.24),(3.28). Then Ytφ = ρt (X − tT φtr s dWs ), t ≤ T , and the strategy q tr −1 ⊥ Π⊥ t (Zt ) At Πt (Zt )

φ¯t =

q

h2t − ξttr At ξt

(3.29)

At ξt + Πt (Zt )

¯

is in Φ and satisfies Ytφ = ess inf Ytφ = Yt for any t ∈ [0, T ], that is φ∈Φ



πtu (X) = ess inf ρt X − φ∈Φ

Z t

T





c φtr s dWs = ρt X −

Z t

T

φ¯ c φ¯tr s dWs = Yt .



¯

Moreover, the tracking error Rφ (X) is a supermartingale under all measures Q ∈ P ngd and a martingale under the measure Qλ ∈ P ngd with Girsanov kernel ¯ −1/2 A−1 Zt − φ¯t , λt := ht (Zt − φ¯t )A−1 t t (Zt − φt ) 



t ∈ [0, T ].

One could interpret the dynamics of the no-good-deal valuation (3.24) as follows. By dYt =: ct + Π⊥ (Zt )dW c (cf. Section 3.1.1) it −at dt + Πt (Zt )ξt dt + Zt dWt = −at dt + Πt (Zt )dW t c , that is dynamically decomposes into a hedgeable part Πt (Zt )(ξt dt + dWt ) = Πt (Zt )dW c spanned by tradeable assets, an orthogonal part Π⊥ t (Zt )dW , being a martingale under P (and b and a remaining part being an absolutely continuous (finite variation) process whose rate Q), at ≥ 0 may be seen as a premium inherent to the upper good deal bound to compensate the seller of the claim for non-tradeable risk. Note that a > 0 on {(ω, t) : Π⊥ (Zt ) 6= 0}. The summands in the expression (3.29) for the strategy φ¯ play different roles from the perspective of hedging. The second summand is a non-speculative component that hedges locally tradeable risk by replication, while the first is a speculative component that compensates (“hedges”) for unspanned non-tradeable risk by taking favorable bets on the market price of risk. Clearly, good deal bounds fit into the rich theory of g-expectations and market-consistent risk measures (cf. [BE09] and more references therein). See [Lei07] for closely related ideas about instantaneous measurement of risk.

Section 3.2. Dynamic good-deal hedging

3.2.2

Page 113

Examples for good-deal valuation and hedging with closed-form solutions

Explicit formulas, if available, facilitate intuition and enable fast computation of valuations, hedges and comparative statics. To this end, several concrete case studies are provided, starting with European options with monotone payoff profiles (e.g. call options) on non-traded assets in a multidimensional model of Black-Scholes type, in which tradeable assets only permit for partial hedging. In parallel to [CT14, Proposition 3, Section 5.3] and [BY08], who employ SDE respectively PDE methods, this demonstrates how previous BSDE analysis can be applied in concrete case studies and we contribute some slight generalizations as well (e.g. higher dimensions, ellipsoidal constraints). As a further example, we contribute new explicit formulas for an option to exchange (geometric averages of) non-traded assets into traded assets. As before, the no-good-deal approach here gives rise to a familiar option pricing formula (by Margrabe) but suitable adjustments of parameter inputs are required, showing the difference to a simple no-arbitrage valuation approach that uses only one (given) single risk neutral measure. A further example derives semi-explicit good-deal solutions for the stochastic volatility model by Heston, for no-good-deal constraints on market prices of (unspanned) stochastic volatility risk which impose an interval range on the mean reversion level of the stochastic variance process under any valuation measure Q ∈ Qngd . Technically, this corresponds to imposing bounds on the instantaneous Sharpe ratio which are inversely proportional to the stochastic volatility. This is different to a related result by [BL09], in that their example imposes no good deal constraints in terms of bounds on simultaneous changes in the level of mean-reversion combined with opposite changes in reversion speed. We emphasize that, in addition to valuation formulas, all our examples provide explicit results for good-deal hedging strategies as well. Detailed derivations of the formulas in Sections 3.2.2-3.2.2 are given in Appendix 3.4 Closed-form formulas for options in a generalized Black-Scholes model The market information F = (Ft )t≤T is generated by an n-dimensional P -Brownian motion W := (W 1 , . . . , W n )tr with W S = (W 1 , . . . , W d )tr , d < n for n, d ∈ N, and is augmented by null-sets. The financial market consists of d ≤ n (incomplete if d < n) stocks with (discounted) prices S = (S k )dk=1 and further n − d non-traded assets with values H = (H l )n−d l=1 . We consider e b a risk neutral model (P = Q ∈ M , ξ = 0) where the processes S and H evolve as dSt = diag(St )σ S dWtS

and dHt = diag(Ht ) γdt + βdWt , 

t ∈ [0, T ],

S) d×d invertible, with S0 ∈ (0, ∞)d , H0 ∈ (0, ∞)n−d , constant coefficients σ S = (σki k,i ∈ R β = (βli )l,i ∈ R(n−d)×n and γ ∈ Rn−d . The volatility matrix of S is σ := (σ S , 0) ∈ Rd×n and is clearly of maximal rank d ≤ n. For z ∈ Rn , we have Π(z) = (z 1 , . . . , z d , 0, . . . , 0)tr ∈ Rn

and Π⊥ (z) =



0, . . . , 0, z d+1 , . . . , z n

tr

∈ Rn . We assume the ellipsoidal framework of

Section 3.2. Dynamic good-deal hedging

Page 114

Section 3.2.1, with h ≡ const > 0 and A ≡ diag(a), with a ∈ (0, ∞)n . Clearly A satisfies the assumption (3.22). By convention,set 0/0  = 0. From Theorem 3.15 we know that ¯ Q u ¯ ¯ πt (X) = Yt = Et [X] with dQ/dP = E λ · W where ¯t = h λ

n X

(Zti )2 /ai

−1/2

(0, . . . , 0, Ztd+1 /ad+1 , . . . , Ztn /an )tr ,

i=d+1

for (Y, Z) solving the classical BSDE n X

−dYt = h

(Zti )2 /ai

1/2

dt − Zttr dWt , t ≤ T,

and YT = X.

(3.30)

i=d+1

By Theorem 3.17 the good-deal hedging strategy is φ¯t = Π(Zt ), t ≤ T . Define the 1/d  Qd ˜ t = Qn−d Htl 1/(n−d) . Then one can rewrite geometric averages S˜t = Stk and H k=1 l=1     ˜ 2 t , where ˜t = H ˜ 0 exp β˜tr Wt + γ˜ − 1 |β| S˜t = S˜0 exp σ ˜ tr WtS + µ ˜ − 12 |˜ σ |2 t and H 2 ˜ 2 − 1 |β|2 , σ |2 − 1 |σ S |2 , β˜ = 1 β tr 1 and γ˜ := 1 γ tr 1+ 1 |β| σ ˜ := 1 (σ S )tr 1, µ ˜ := 1 |˜ d

2

2d

n−d

with 1 = (1, . . . , 1)tr . We treat the following two examples.

n−d

2

2(n−d)

˜ T ) in European option on non-traded assets: Consider a European option X = G(H 2 ˜ L on the geometric average H, where x 7→ G(x) is a non-decreasing measurable payoff function of polynomial growth in x±1 , i.e. |G(x)| ≤ k(1 + xn + x−n ) for all x > 0, ¯ is a constant process given by λ ¯t = for some k > 0 and n ∈ N. We claim that λ −1/2 Pn tr 2 ˜ ˜ ˜ h (0, . . . , 0, βd+1 /ad+1 , . . . , βn /an ) . Let (Y, Z) be the standard solution i=d+1 βi /ai ¯ For λ ¯ constant, the Feynman-Kac formula yields to the linear classical BSDE (3.7) with λ = λ.  ¯ ˜ T )] = u(t, H ˜ t ∂x u(t, H ˜ t ) and Zt = H ˜ t )β˜ for a function u ∈ C 1,2 [0, T ) × (0, ∞) Yt = EtQ [G(H solution to a Black-Scholes type PDE (after coordinate transformations that reduce the PDE into the heat equation using [KS06, Section 4.3]). Since G is non-decreasing, ∂x u ≥ 0 holds. 1/2 i 2 ¯ tr Zt = h Pn ¯ is Because of this one actually has λ , t ∈ [0, T ]. Hence λ t i=d+1 (Zt ) /ai ¯ ˜ T )]. The process H ˜ satisfies indeed the constant process given above, and πtu (X) = EtQ [G(H 1/2  Pn 1 ˜2 α t tr 2 ˜ ˜ + ˜ ˜ ¯ ¯ Ht = H0 e exp β Wt − 2 |β| t , t ∈ [0, T ], where α± := γ˜ ± h and W i=d+1 βi /ai ¯ is an n-dimensional Q-Brownian motion. Specifically for G(x) := (x − K)+ , X is a call option ˜ with strike K and maturity T . The upper good-deal bound is given for t ∈ [0, T ] by a on H Black-Scholes type formula (with “vol” abbreviating volatility) ˜ t eα+ (T −t) − KN (d− ) πtu (X) = N (d+ )H ˜ , (3.31) ˜ t , strike: Ke−α+ (T −t) , vol: |β| =eα+ (T −t) ∗ B/S-call-price time: t, spot: H √  ˜ 2 (T − t) |β| ˜ T − t−1 and N is the cdf of the ˜ t /K + α+ ± 1 |β| where d± := ln H 2 standard normal law. Analogously, the lower good-deal bound turns out as ˜ . ˜ t , strike: Ke−α− (T −t) , vol: |β| πtl (X) = eα− (T −t) ∗ B/S-call-price time: t, spot: H

Section 3.2. Dynamic good-deal hedging

Page 115

The difference between the good-deal valuation formulas above and the standard Black-Scholes b b for a call option (H ˜ T − K)+ formula for risk-neutral valuation EtQ [X] under measure P = Q ˜ t , which reduce to the risk-neutral shows in the factors eα± (T −t) multiplying the spot price H P case eγ˜(T −t) if h = 0, i.e. α± = γ˜ . The difference α± − γ˜ = ±h( n β˜2 /ai )1/2 when i=d+1 i at π·u (X)

h > 0 translates into an additional premium an option trader (selling or buying at π·l (X)) would require, if using the no-good-deal approach instead of the arbitrage-free valuation b (being an element of Qngd ). The good-deal hedging under a given risk neutral measure P = Q strategy for the seller of X in terms of parametrizations of Section 3.1.1 is  ˜ t β˜1 , . . . , β˜d , 0, . . . , 0 tr , t ∈ [0, T ], φ¯t = eα+ (T −t) N (d+ )H

(3.32)

b for the call option which coincides with the delta hedging strategy (as computed under P = Q) + ˜ T − K) only if α+ is zero and the risky asset H ˜ is tradeable. The hedging strategy of the (H buyer is derived analogously.

Exchange option of traded and non-traded assets: Consider an European option to ˜ at maturity T with payoff X = exchange the traded asset S˜ for the non-traded asset H ¯ ˜ T − S˜T )+ ∈ L2 . The upper bound π u (X) = E Q [X] can be explicitly derived (see Appendix) (H t t using arguments from the previous example in combination with a change of num´eraire. We thereby obtain a Margrabe type formula ˜ t eα+ (T −t) − N (d− )S˜t eµ˜(T −t) πtu (X) = N (d+ )H (3.33)  α (T −t) µ ˜ (T −t) ˜ te + = B/S-call-price time: t, spot: H , strike: S˜t e , vol: δ , −1   √ 2 ˜ t /S˜t + (α+ + µ where d± := ln H ˜ ± δ2 )(T − t) δ T − t . Analogously, the corresponding lower good-deal bound is

˜ t eα− (T −t) , strike: S˜t eµ˜(T −t) , vol: δ . πtl (X) = B/S-call-price time: t, spot: H 

The good-deal hedging strategy φ¯t for the seller of the exchange option equals  tr ˜ t eα+ (T −t) β˜1 , . . . , β˜d , 0, . . . , 0 tr − N (d− )S˜t eµ˜(T −t) σ N (d+ )H ˜1 , . . . , σ ˜d , 0, . . . , 0 .

Again, the difference between the good-deal valuation formula and the classical Margrabe b for formula, as computed by standard no-arbitrage valuation under risk neutral measure P = Q, + α (T −t) ˜ T − S˜T ) shows by the presence of the factors e ± the exchange option (H involving the Pn 2 1/2 ˜ term ±h( i=d+1 βi /ai ) , which depends only on the parameters A and h for no-good-deal restrictions.

Section 3.2. Dynamic good-deal hedging

Page 116

Computational results by Monte Carlo To demonstrate that good-deal bounds and hedging strategies can be computed numerically in moderately high dimensions by generic simulation methods available for classical BSDE, we apply the (generic) multilevel Monte Carlo algorithm from [BT14] (that builds on [GT15]) to approximate the solution (Y, Z) of the BSDE (3.30) in dimension n = 4, and compare with ˜ T − S˜T )+ . Using parameters the known analytical solution for the exchange option X := (H d = 2, T = 1 and !

H0 = β=

!

1 , 1

0.3 0.5

S0 =

0.4 0.7

0.2 0.3

1 , 1

σS =

0.5 0

!

0.2 , 0.4

γ = (0.1, 0.3)tr ,

!

0.5 , 0.4

h = 0.3,

and A = diag(0.5, 0.65, 0.8, 0.95),

we compare the approximate values at time t = 0 to the known theoretical values obtained from Section 3.2.2. The exact value of the good-deal bound at time t = 0 according to the formula (3.33) is then π0u (X) = 0.5494, up to four digits, while for the hedging strategy it is φ¯0 = (0.3049, 0.4440, 0, 0), the exact value of Z0 being (0.3049, 0.4440, 0.2792, 0.5025). We use a 4-level algorithm on an equidistant time grid with N = 24 steps, a number of sample paths M = 3 × 106 and with K = 504 regression functions, being indicator functions on a hypercube partition of R4 , the state space of the forward process (S, H). Table 3.1 provides the numerical simulation results, summarized by the approximation means for the good-deal bound and the hedging strategy at time 0, the empirical root-mean-square errors (RMSE) computed coordinate-wise and the corresponding relative values (Rel.RMSE), based on 80 independent simulation runs. Simulation in Matlab for one run took 153sec on a core-i7 cpu laptop, showing relative errors (in terms of maximal coordinates in Rel.RMSE) of about 0.07% for valuation and 0.34% for hedging. Y0 approx

Z0 approx

φ¯0 approx

Mean

0.5499

(0.3052, 0.4462, 0.2852, 0.5137)

(0.3052, 0.4462, 0, 0)

RMSE

10−4 × 4

10−4 × (10, 13, 12, 13)

10−4 × (10, 13, 0, 0)

Rel.RMSE

10−4 × 7

10−4 × (34, 29, 41, 27)

10−4 × (34, 29, 0, 0)

Table 3.1: Mean and (relative) root-mean-square errors of approximations

Section 3.2. Dynamic good-deal hedging

Page 117

Semi-explicit formulas in the Heston stochastic volatility model The market information is generated by a 2-dimensional P -Brownian motion W = (W S , W ν ), and is augmented by null-sets. We are going to consider a European put option X = (K − ST )+ on S with strike K in the Heston model q  √ √ a dSt = St νt dWtS and dνt = b( − νt )dt + β νt ρdWtS + 1 − ρ2 dWtν , t ≤ T, b b with S0 , ν0 > 0, a, b, β > 0 and that is specified directly under a risk neutral measure P = Q, ρ ∈ (−1, 1). Here the variance process ν is a CIR process with b representing the mean-reversion speed, a/b the mean-reversion level and β/2 the volatility of the variance. Assume that the condition β 2 ≤ 2a is satisfied, such that by the Feller’s test for explosions (cf. [KS06, Theorem 5.5.29]) applied to the process ln(ν) the variance process ν is strictly positive. In the sequel we refer to this condition (i.e. β 2 ≤ 2a) for a CIR process as the Feller condition. The equivalent local martingale measures Q ∈ Me in this model are specified by Girsanov kernels λ such that dQ/dP = E(λ · W ν ) is a uniformly integrable martingale. Indeed, we parametrize the pricing measures only by the second component of their Girsanov kernels (i.e. with respect to W ν ) since the first component is always zero. We consider the no-good-deal constraint correspondence q

Ct (ω) = x ∈ R2 : |x| ≤ ε/ νt (ω) 



(t, ω) ∈ [0, T ] × Ω,

(3.34)

for a constant ε > 0. One observes that C is standard with 0 ∈ C, non-uniformly bounded √ and satisfies (3.9) for R = ε/ ν (since ν > 0 is continuous). Hence good-deal valuation results for uniformly bounded correspondences may not apply. Using [CFY05], we can obtain a convenient Heston-type formula (semi-explicit, computation requiring only 1-dim. integration) for the good-deal bound of the put option X = (K − ST )+ , πtu (X)

q

= Heston-put-price(time: t, a ¯ := a + βε 1 − ρ2 , b, β),

(3.35)

just like the ordinary Heston put price, associated to parameters (t, a, b, β), but where the p parameter a has to be adjusted to a ¯ := a + βε 1 − ρ2 . The formula for the lower bound πtl (X) p is similar, but with a ¯ replaced by a := a − βε 1 − ρ2 , for which the Feller condition β 2 ≤ 2a ¯ is still satisfied if ε ≤ 12 β −1 (2a − β 2 )(1 − ρ2 )−1/2 . In particular, πtu (X) = EtQ [X] holds with  √ ¯ dQ/dP = E (ε/ ν) · W ν . By Corollary 3.10 this yields Y¯ = π·u (X) for the minimal solution ¯ ∈ S ∞ × H2 of the BSDE (Y¯ , Z) ε −dYt = √ Zt2 − Zttr dWt , t ∈ [0, T ], νt

YT = (K − ST )+ .

(3.36)

The (seller’s) good-deal hedging strategy φ¯ is given by the semi-explicit formula √ βρ φ¯t = St νt ∆t + Vt , 2

(3.37)

Section 3.2. Dynamic good-deal hedging

Page 118

where ∆t and Vt denote the delta and the vega of the put option at time t in the Heston model with parameters (¯ a, b, β). Derivations are provided in Appendix 3.4. We note that (3.37) coincides (cf. [PSHE09]) with the risk-minimizing strategy (in the sense of [Sch01]) for the put ¯ in a Heston model, not with respect to the probability P but with respect to the measure Q (derived just before) under which also Heston dynamics but with modified parameters prevail. This shows, how the strategy (3.37) differs from the standard risk minimizing strategy under P (as in [PSHE09, HPS01]). Good-deal valuation bounds for a put option in the Heston model are thus given by a Heston type formula but for a mean-reversion level increased by p βε 1 − ρ2 /b > 0. Similar to earlier examples, this difference constitutes an increase in the premium that an issuer selling at π·u (X) would require according to good-deal valuation, in b comparison to a standard arbitrage free valuation under one given risk neutral measure P = Q, when S is the only risky asset available for hedging and stochastic volatility risk is otherwise taken to be unspanned. Figures 3.1,3.2,3.3 graphically illustrate this, showing the good-deal valuations π0u (X), π0l (X) (at t = 0) for a long-dated put option with maturity T = 10 in relation to the underlying S0 , to the correlation coefficient ρ and to the no-good-deal constraint parameter ε (for bound √ on optimal growth rate h = ε/ ν) respectively. Other global parameters are K = 100, a = 0.12, b = 3, β = 0.3, ν0 = 0.04. Computations of the Heston formula have been done in Matlab following the algorithm of [KJ05]. Figure 3.1 is a plot of π0u (X), π0l (X) as function of initial stock price S0 for values of ε in {0.15, 0.25}. Similarly Figure 3.2 provides a plot illustrating the variation with ρ for ε ∈ {0.1, 0.2}, while Figure 3.3 illustrates the dependence on ε. The largest value 0.35 for ε in Figure 3.3 has been chosen as the maximal one allowing p for the Feller condition β 2 ≤ 2a = a − βε 1 − ρ2 for the lower bound π0l (X) to be satisfied, i.e. ε ≤ 12 β −1 (2a − β 2 )(1 − ρ2 )−1/2 ≈ 0.35 for the chosen parameters. Because the values of ε are close to zero, the lines in Figure 3.3 may look straight at the first impression, but by having a closer look the reader can convince himself that the lines are indeed not straight as expected. We could have plotted the upper bound π0u (X) for larger values of ε, but we simply chose to use on the same plot the same range of ε as that for the lower bound π0l (X). The standard Heston price computed directly under a given risk neutral (minimal martingale) b (i.e. for ε = 0) lies between the upper and lower good-deal bounds, whose measure P = Q spread increases with ε > 0. The monotonicity in ε is intuitively obvious since as  increases, the correspondence C maps to larger sets, yielding weaker no-good-deal constraints which then imply wider good-deal valuation bounds. That the bounds in Figure 3.2 coincide for perfect correlation ρ ∈ {−1, 1} is also intuitively clear. Indeed since for |ρ| = 1 volatility risk is entirely spanned by the tradeable asset, then the former can be perfectly hedged such that the Heston model becomes complete and π0u (X) = π0l (X) = E Qb [X] holds for all contingent claims X.

Section 3.2. Dynamic good-deal hedging

Page 119

100

Heston price under Min-Mart-Measure: ǫ=0

90

upper & lower bounds for ǫ=0.15 upper & lower bounds for ǫ=0.35

80

70

π u0 , π l0

60

50

40

30

20

10

0 0

50

100

150

S0

Figure 3.1: Dependence of π0u (X), π0l (X) on S0 for ρ = −0.7 and T = 10.

42

40

Heston price under Min-Mart-Measure: ǫ=0 upper & lower bounds for ǫ=0.1 upper and lower bounds for ǫ=0.2

38

36

π u0 , π l0

34

32

30

28

26

24

22 -1

-0.8

-0.6

-0.4

-0.2

ρ

0

0.2

0.4

0.6

0.8

1

Figure 3.2: Dependence of π0u (X), π0l (X) on ρ for S0 = 100 and T = 20.

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 120

26

upper bound: π u0(X)

24

lower bound: π l0(X)

π u0 , π l0

22

20

18

16

14

12 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

ǫ

Figure 3.3: Dependence of π0u (X), π0l (X) on ε for S0 = 100, ρ = −0.7 and T = 10.

3.3

Good-deal valuation and hedging under model uncertainty

In preceding sections, good-deal bounds and hedging strategies have been described by classical BSDEs under the probability measure P , expressing the objects of interest in terms of the market price of risk ξ with respect to P . In reality, the objective real world probability measure is not precisely known, hence there is ambiguity about the market price of risk. To include model uncertainty (ambiguity) into the analysis, we follow a multiple priors approach in spirit of [GS89, CE02, ES03], by specifying a confidence region of reference probability measures {P θ : θ ∈ Θ} (multiple priors, interpreted as potential real world probabilities of equal right), centered around some measure P0 . In practice, an investor facing model uncertainty may first extract an estimate P0 for the true but uncertain P from data, but then consider a class R of potential reference measures in some confidence region around P0 to acknowledge the statistical uncertainty of estimation. Starting point for good-deal valuation approach under uncertainty is then to associate to each model P θ its own family of (a-priori) no-good-deal measures Qngd (P θ ) (resp. P ngd (P θ )). A robust worst-case approach requires the seller of a derivative to consider ¯ the (worst-case) model P θ that provides the largest upper good-deal valuation bound, to be conservative against model misspecification (see (3.58)). Such leads to wider good-deal bounds, corresponding to a larger overall set of no-good-deal measures under uncertainty. Notably, it will simultaneously also give rise to a suitable robust notion of good-deal hedging, which is uniform with respect to all P θ , by means of a saddle point result that ensures a minmax identity

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 121

(see Theorem 3.30). We associate to each model P θ a correspondence C θ that defines the set of no-good-deal measures in this model. The aggregate set of no-good-deal measures will be e which incorporates also the uncertainty. Technically, described then by single correspondence C, this makes is possible to apply analysis obtained in the framework of previous sections of this chapter, with P0 taking the role of P .

3.3.1

Model uncertainty framework

Let (Ω, F, P0 , F) be a probability space with a usual filtration F = (Ft )t≤T generated by an n-dimensional P0 -Brownian motion W 0 . We assume that all reference measures P θ are equivalent to P0 with corresponding Girsanov kernels θ evolving in some given confidence region Θ. More precisely, we define

n





o

R := P θ ∼ P0 dP θ /dP0 = E θ · W 0 , with θ predictable and θ ∈ Θ , where Θ : [0, T ] × Ω Rn is a standard correspondence satisfying Assumption 3.3 and 0 ∈ Θ, hence P0 ∈ R 6= ∅. A similar framework has been considered for example in [CE02, Que04] for solving the robust utility maximization problem under Knightian uncertainty about drift coefficients. We do write θ ∈ Θ for θ being a predictable selection of Θ. The financial market consists of d ≤ n tradeable risky assets whose discounted prices (S i )di=1 under P θ (for θ ∈ Θ) evolve as Itˆo processes, solving the SDEs cθ, dSt = diag(St )σt (ξtθ dt + dWtθ ) =: diag(St )σt dW t

(3.38)

t ≤ T,

with S0 ∈ (0, ∞)d , for Rn -valued predictable ξ θ and Rd×n -valued predictable volatility σ of R full rank, and W θ := W 0 − 0· θs ds a P θ -Brownian motion. Noting that market prices of risk, ξtθ and ξt0 , canonically take values in Im σttr , we assume that market prices of risk ξ θ (under P θ for θ ∈ Θ) have the form (3.39)

ξtθ = ξt0 + Πt (θt ) ∈ Im σttr , t ∈ [0, T ],

and that ξ 0 is bounded. By (3.39), the solutions of the SDEs (3.38) coincide P0 -a.s. for all θ ∈ Θ. The process ξ θ (for θ ∈ Θ) is the market price of risk in the model P θ and is also bounded (since ξ 0 is bounded and Θ satisfies Assumption 3.3). Hence, the minimal b θ with respect to P θ is dQ b θ = E(−ξ θ · W θ )dP θ . In addition martingale measure [Sch01] Q  R b θ = E Π⊥ (θ) · W c 0 dQ b 0 and W cθ = W c 0 − · Π⊥ (θt )dt, for all θ ∈ Θ. We recall from dQ 0 t Section 3.1.1 how dynamic trading strategies are defined and re-parametrized in terms of c 0 . The set of permitted trading strategies is integrands (φi )di=1 with respect to W n



Φ := φ φ is predictable, φ ∈ Im σ tr and E P0

hZ 0

T

i

o

|φt |2 dt < ∞ .

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 122

Since φtr Π⊥ (θ) = 0 for θ ∈ Θ, the wealth process V φ of strategy φ ∈ Φ with initial capital V0 c θ = V0 + φ · W c 0 , for all θ ∈ Θ. Let Me (P θ ) := Me (S, P θ ) denote the set is V φ = V0 + φ · W of equivalent local martingale measures for S in the model P θ . Noting P θ ∼ P 0 and recalling Proposition 3.2 one easily obtains e θ e e θ Proposition 3.18.   M (P ) = M (P0 ) for allθ ∈ Θ. In addition, every Q ∈ M (P ) satisfies dQ = E λθ · W θ dP θ and dQ = E λ0 · W 0 dP0 , with λθ = −ξ θ + η θ and λ0 = −ξ 0 + η 0 ,

where Π⊥ (λθ ) = η θ ,

Π⊥ (λ0 ) = η 0 and η θ = η 0 − Π⊥ (θ).

Thus, we simply write Me = Me (S) for the set of equivalent martingale measures.

3.3.2

No-good-deal constraint and good-deal bounds under uncertainty

Let {C θ | θ ∈ Θ} be a family of standard correspondences satisfying −ξ θ ∈ C θ

for all θ ∈ Θ.

(3.40)

In the model P θ , θ ∈ Θ, let the no-good-deal constraint be such that the Girsanov kernels of measures in Me are selections of C θ . The resulting set Qngd (P θ ) of no-good-deal measures is equal to n



o

Q ∼ P θ dQ/dP θ = E λ · W θ , λ predictable, bounded, λ ∈ (−ξ θ + Ker σ) ∩ C θ . 

b θ ∈ Qngd (P θ ) 6= ∅ for all θ ∈ Θ. By Proposition 3.18 holds By (3.40), then Q n



o

Qngd (P θ ) = Q ∼ P0 dQ/dP0 = E λ · W 0 , λ ∈ −ξ 0 + (Ce θ ∩ Ker σ) 

(3.41)

where λ is predictable and bounded, and for all θ ∈ Θ the correspondences Ce θ are given by Ce θ := C θ + ξ θ + Π⊥ (θ) = C θ + ξ 0 + θ.

(3.42)

Following a worst-case approach, we take the (robust) upper good-deal valuation π·u (·) under uncertainty as being the largest of all good-deal bounds π·u,θ (·) over all models P θ , θ ∈ Θ. The respective set Qngd of no-good-deal valuation measures corresponding to π·u (·) can be described in terms of the sets Qngd (P θ ), θ ∈ Θ. At first, one might guess that Qngd should be the union of all Qngd (P θ ). However, to have m-stability and convexity of Qngd for good dynamic properties of the resulting good-deal bounds (as in Lemma 3.1), one has to define Qngd as the smallest m-stable and convex set containing all Qngd (P θ ), θ ∈ Θ. Definition 3.19. Qngd is the smallest m-stable convex subset of Me containing all Qngd (P θ ), θ ∈ Θ. For sufficiently integrable claims X (e.g. in L∞ ), the worst-case upper good-deal bound under uncertainty is πtu (X) := ess supQ∈Qngd EtQ [X].

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 123

We characterize the set Qngd from Definition 3.19 using a suitable single correspondence Ce which is derived from all C θ , θ ∈ Θ. To this end, we impose the Assumption 3.20. The correspondence with values compact-valued and predictable.

eθ θ∈Θ Ct (ω),

S

(t, ω) ∈ [0, T ] × Ω, is

The theory of measurable correspondences is well-developed for closed-valued correspondences (see [Roc76]). Assumption 3.20 ensures closed-valuedness and predictability of Ce for the proposition below. If all C θ (θ ∈ Θ) are equal to some given C 0 , as in the following example, such an assumption will automatically hold in the setting required for Section 3.3.4, where Ce θ (θ ∈ Θ) are ellipsoidal. Example 3.21. For a standard correspondence C 0 with ξ θ ∈ C 0 , θ ∈ Θ, let C θ := C 0 , θ ∈ Θ. S Then Ce θ = C 0 + ξ 0 + θ and θ∈Θ Ce θ = C 0 + ξ 0 + Θ satisfies Assumption 3.20. Proposition 3.22. Let Assumption 3.20 hold. Then Qngd equals n



o

Q ∼ P0 dQ/dP0 = E λ · W 0 , λ = −ξ 0 + η predictable, bounded, η ∈ Ce , 

(3.43)

S  for the standard correspondence Cet (ω) := Ker σt (ω) ∩ Conv θ∈Θ Cetθ (ω) .

Proof. With Assumption 3.20, [Roc76, Theorem 1.M and Proposition 1.H] imply that Ce is standard. Note that Ce is non-empty-valued since −ξ 0 ∈ C 0 and hence 0 ∈ Cet0 (ω)∩Ker σt (ω) ⊂ Cet (ω). Denote by Q the set in (3.43). By definition Cet (ω) ⊂ Ker σt (ω), implying Q ⊆ Me . We first prove that Qngd ⊆ Q. Applying [Del06, Theorem 1] or following the steps of the proof for Lemma 3.1, Part b), one sees that Q is m-stable and convex. By (3.41) and since Cetθ (ω) ∩ Ker σt (ω) ⊆ Cet (ω) for all θ ∈ Θ, then Q contains the union of all Qngd (P θ ), θ ∈ Θ. By definition Qngd is the smallest m-stable convex subset of Me with this property, hence Qngd ⊆ Q. Let us show Q ⊆ Qngd . The L1 -closure of Qngd is an m-stable closed and convex set of measures Q  P0 , and Qngd comprises exactly those elements of its closure that are equivalent to P0 . Closeness and convexity of the closure of Qngd are clear. We now show its m-stability. To this end, let ZT1 , ZT2 be in the closure of Qngd , τ ≤ T be a stopping time   and ZT := Zτ1 ZT2 /Zτ2 I{Zτ2 >0} + Zτ1 I{Zτ2 =0} . There exist ZT1,n n , n ZT2,n n ⊆ Qngd such that ZT1,n → ZT1 and ZT2,n → ZT2 in L1 . By m-stability of Qngd holds ZTn := Zτ1,n ZT2,n /Zτ2,n ∈ Qngd for each n ∈ N. Now E[ZTn ] = 1 for all n ∈ N, and ZTn → ZT in probability as n → ∞. In addition, E[ZT ] = E[Zτ1 ZT2 /Zτ2 I{Zτ2 >0} ] + E[Zτ1 I{Zτ2 =0} ] = E Eτ [ZT2 /Zτ2 ] Zτ1 I{Zτ2 >0} + E[Zτ1 I{Zτ2 =0} ] = E[Zτ1 ] = 1. 



Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 124

By Scheff´e’s lemma one obtains ZTn → ZT in L1 as n → ∞, and m-stability of the closure of Qngd follows. As W 0 is a continuous P0 -martingale with the predictable representation property, it satisfies the hypotheses of [Del06, Theorem 2], implying by Definition 3.19 the existence of a closed-convex-valued predictable correspondence C 1 such that the no-good-deal measure set Qngd is equal to

n





o

Q ∼ P0 dQ/dP0 = E λ · W 0 , λ = −ξ 0 + η predictable, η ∈ C 1 ∩ Ker σ .

To prove the claim, it suffices to show that all predictable selections of Ce are also predictable selections of C 1 ∩ Ker σ. To this end it suffices to show that for all θ ∈ Θ, any predictable selection of Ce θ ∩ Ker σ is a predictable selection of C 1 ∩ Ker σ. Assume the contrary that there exists θ ∈ Θ and a predictable process η such that η ∈ Ce θ ∩ Ker σ and η is not selection  of C 1 ∩ Ker σ. Then E (−ξ 0 + η) · W 0 is in Qngd (P θ ) but not in Qngd , which contradicts Qngd (P θ ) ⊆ Qngd .

Using the characterization of Qngd in Proposition 3.22 we can apply the results of Sections 3.13.2 in order to derive worst-case good-deal bounds and hedging strategies under uncertainty like in the absence of uncertainty, with the center P0 of the set of reference measures R taking the role of P (in Sections 3.1-3.2) and the enlarged correspondence Ce taking the role of C there. Example 3.23. For C θ , θ ∈ Θ, as in Example 3.21 holds Ce = (C 0 + ξ 0 + Θ) ∩ Ker σ and n







Qngd = Q ∼ P0 dQ/dP0 = E λ · W 0 , λ ∈ (−ξ 0 + Ker σ) ∩ (C 0 + Θ)

o

(3.44)

with λ denoting bounded predictable selections, by Proposition 3.22. Moreover the union S ngd (P θ ) is convex, m-stable (cf. Lemma 3.26) and equals Qngd . θ∈Θ Q Remark 3.24. a) Equation (3.43) shows, how the good-deal valuation and hedging problem under model uncertainty can technically be embedded into the mathematical framework of Sections 3.1-3.2 without uncertainty, by considering an enlarged no-good-deal constraint correspondence C as Conv(∪θ∈Θ (C θ +θ)) in (3.6) with P0 taking the role of P . In Example 3.23, (3.44), it simply means to take C as C 0 + Θ. b) Typical examples for good-deal constraints are radial, i.e. C 0 is a ball. This case is predominant in the literature and justified from a finance point of view by ensuring a constant bound on instantaneous Sharpe ratios (or growth rates). But typical examples for uncertainty (ambiguity) constraints Θ can well be non-radial (see [CE02, ES03]). For instance, Θ may arise from a confidence region for some unknown drift parameters in a multivariate (log-)normal model; such would in general be ellipsoidal but not radial, and the sum C 0 + Θ can even be

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 125

non-ellipsoidal. To offer a suitable framework for such and other examples, Section 3.1 treats abstract correspondences. A constructive method to solve for such a typical parametrization of C 0 + Θ is described in Remark 3.31.

3.3.3

Robust approach to good-deal hedging under model uncertainty

ngd θ As in Sectionn3.2 (cf. (3.15) and the definition   of Q (P )), we define for θ ∈ o Θ the set θ θ θ θ ngd θ P (P ) := Q ∼ P | dQ/dP = E λ · W , λ ∈ C predictable, bounded in order to

introduce a robust notion of good-deal hedging. Let P ngd denote the smallest m-stable convex set of measures Q ∼ P0 containing all P ngd (P θ ), θ ∈ Θ. Then ρt (X) := ess sup EtQ [X],

t ∈ [0, T ], X ∈ L2 (P0 ),

Q∈P ngd

defines a time-consistent dynamic coherent risk measure by Lemma 3.1. Like in Section 3.2, the good-deal hedging problem under uncertainty is posed as a minimization problem (3.45) of a-priori risk measures ρ of hedging errors: for a contingent claim X, find a strategy φ∗ ∈ Φ such that for all t ∈ [0, T ] holds 

πtu (X) = ρt X −

T

Z t





c 0 = ess inf ρt X − φ∗s tr dW s φ∈Φ

Z t

T



c0 φtr s dWs .

(3.45)

The good-deal hedging strategy under uncertainty is defined as this minimizer (if it exists) φ∗ ∈ Φ. For X ∈ L2 (P0 ), one can prove (as in Proposition 3.12) that the tracking error ∗ Rφ (X) (defined as in (3.18)) of the strategy φ∗ is a supermartingale under every measure in P ngd : Proposition 3.25. For X ∈ L2 (P0 ), let φ∗ be the strategy solving (3.45). Then the tracking ∗ error Rφ (X) of this strategy is a Q-supermartingale for all Q ∈ P ngd . A strategy solving the good-deal hedging problem under uncertainty and whose tracking error satisfies the supermartingale property under all measures in P ngd (as in Proposition 3.25) will be qualified as robust with respect to uncertainty. Note that this is a different notion of robustness compared to the one in Remark 2.12, because the supermartingale property has S to hold for measures in P ngd (P θ ) uniformly for all models P θ ∈ R (since θ∈Θ P ngd (P θ ) is a subset of P ngd ). More concrete results under uncertainty will be derived next under additional conditions.

3.3.4

Hedging under model uncertainty for ellipsoidal good-deal constraints

In this section we consider ellipsoidal good-deal constraints. To this end, let n



o

Ct0 (ω) = x ∈ Rn xtr At (ω)x ≤ h2t (ω) ,

(t, ω) ∈ [0, T ] × Ω,

(3.46)

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 126

where A is a uniformly elliptic and predictable matrix-valued process, and h some positive bounded and predictable process. We assume that A satisfies the separability condition (3.22) with respect to σ. Let Θ be an arbitrary standard correspondence satisfying the uniform boundedness Assumption 3.3 and 0 ∈ Θ. As in Example 3.21, we let C θ := C 0 , for all θ ∈ Θ, yielding by (3.42) that tr



n

Cetθ ∩ Ker σt = x ∈ Rn xtr At x ≤ h2t − ξtθ At ξtθ

o\

Ker σt + Π⊥ t (θt ).

Clearly, C θ is standard and satisfies Assumption 3.3 for θ ∈ Θ. Similarly to (3.27), to derive explicit BSDE formulations for solving the hedging problem we will assume √ (3.47) |ξ θ | < h α0 for all θ ∈ Θ, where the process α0 is the constant of ellipticity of A−1 as in Lemma 3.16. Recall that, thanks to Lemma 3.16, the inequality (3.47) implies in particular that −ξ θ ∈ C 0 , θ ∈ Θ; hence (3.40) holds and the correspondences Ce θ ∩ Ker σ are standard, θ ∈ Θ. Note that condition (3.47) ensures applicability of Lemma 3.35 in our current setup for any model P θ . Since C θ is equal to C 0 and satisfies Assumption 3.3, one has n



o

P ngd (P θ ) = Q ∼ P0 dQ/dP0 = E λ · W 0 , λ predictable, λ ∈ C 0 + θ . 

(3.48)

The following lemma is proven in the Appendix. Lemma 3.26. 2. The set

1. The set

S

ngd (P θ ) θ∈Θ Q

S

θ∈Θ P

ngd (P θ )

is m-stable, convex and equal to P ngd .

is m-stable, convex and equal to Qngd .

Thanks to Lemma 3.26, the dynamic risk measure ρ satisfies for X ∈ L2 (P0 ) ρt (X) := ess sup EtQ [X] = ess sup ρθt (X), Q∈P ngd

t ∈ [0, T ],

θ∈Θ

with ρθt (X) := ess supQ∈P ngd (P θ ) EtQ [X]. The worst-case upper good-deal bound πtu (X) for X ∈ L2 (P0 ) rewrites from Definition 3.19 as πtu (X) := ess sup ess sup EtQ [X] = ess sup πtu,θ (X), t ∈ [0, T ], θ∈Θ

Q∈Qngd (P θ )

θ∈Θ

(3.49)

where πtu,θ (X) = ess supQ∈Qngd (P θ ) EtQ [X]. The corresponding lower bound π·l (X) is obtained via π·l (X) = −π·u (−X). For a worst-case approach to uncertainty we will investigate valuation of claims according to π·u (·) and hedging with the optimal trading strategy solution to (3.45). We employ results from Section 3.2.1 (under P = P θ ) to characterize π·u,θ (X) as well as the associated hedging strategies φ¯θ in Φ. For θ ∈ Θ and φ ∈ Φ let us consider the classical BSDEs −dYt = f φ,θ (t, Zt )dt − Zttr dWt0 , −dYt = f θ (t, Zt )dt − Zttr dWt0 ,

t ≤ T, t ≤ T,

YT = X YT = X,

and

(3.50) (3.51)

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 127

with generators tr

f φ,θ (t, z) = θttr (z − φt ) − ξt0 φt + ht (z − φt )tr A−1 t (z − φt )

1/2

tr ⊥ 0 tr Π⊥ t (θt ) Πt (z) − ξt Πt (z) 1/2 ⊥ tr −1 ⊥ 1/2 tr h2t − ξtθ At ξtθ Πt (z) At Πt (z) .

θ

f (t, z) = +

(3.52) (3.53)

It is straightforward to derive the BSDE descriptions for π·u,θ (X) and ρθ· (X) stated in the subsequent proposition. The proof is analogous to that for Theorem 3.17, using (3.47) instead of (3.27), replacing P by P θ and changing measure from P θ to P0 . Proposition 3.27. Assume (3.22) and (3.47) hold. For X ∈ L2 (P0 ), θ ∈ Θ and φ ∈ Φ, let (Y φ,θ , Z φ,θ ) and (Y θ , Z θ ) be the standard solutions to the BSDEs (3.50) and (3.51) ¯θ ¯ θ ∈ Qngd (P θ ) given by respectively. Then πtu,θ (X) = Ytθ = EtQ [X], t ∈ [0, T ], holds with Q  ¯ θ /dP0 = E (−ξ 0 + η¯θ ) · W 0 for dQ tr

η¯tθ = h2t − ξtθ At ξtθ 

Moreover Ytφ,θ = ρθt X −

1/2

tr

−1/2

−1 ⊥ θ θ Π⊥ t (Zt ) At Πt (Zt )

⊥ θ ⊥ A−1 t Πt (Zt ) + Πt (θt ).



RT

c0 ¯θ φtr s dWs holds, and the strategy φ (in Φ)

t

−1/2 tr θ tr −1 ⊥ θ 1/2 2 φ¯θt := Πt (Ztθ ) + Π⊥ ht − ξtθ At ξtθ At ξtθ t (Zt ) At Πt (Zt )

satisfies

πtu,θ (X)

=

ρθt



X−

Z

T

t





c 0 = ess inf ρθ X − (φ¯θs )tr dW s t

Z

φ∈Φ

t

T



c0 φtr s dWs .

By Proposition 3.27, we can write π·u (X) from (3.49) as πtu (X)

= ess sup θ∈Θ

ess inf ρθt φ∈Φ



X−

Z t

T



c0 φtr s dWs ,

t ∈ [0, T ].

(3.54)

This permits to describe π·u (X) and the associated hedging strategy φ¯ in the next theorem by the solution to the classical BSDE −dYt = f (t, Zt )dt − Zttr dWt0 , t ≤ T

and

YT = X,

(3.55)

with generator f (t, Zt ) := ess supθ∈Θ f θ (t, Zt ), for f θ given in (3.53). The theorem moreover ¯ identifies by θ¯ the worst-case model P θ ∈ R. Theorem 3.28. Assume (3.22) and (3.47) hold. For X ∈ L2 (P0 ), let (Y, Z) be the standard ¯ solution to the BSDE (3.55). Then there exists a unique predictable selection θ¯ := θ(X) of Θ θ ¯ satisfying θt = argmaxθ∈Θ f (t, Zt ) such that for all t ∈ [0, T ] ¯



πtu (X) = ρθt X −

Z t

T

u,θ¯ c0 φ¯tr s dWs = πt (X) = Yt



(3.56)

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 128

¯ holds with φ¯ = (φ¯t )t∈[0,T ] := φ¯θ (X) ∈ Φ given by tr

1/2 2 ¯ ¯−1/2 ¯ tr −1 ⊥ φ¯t = Πt (Zt ) + Π⊥ ht − ξtθ At ξtθ At ξtθ ]. t (Zt ) At Πt (Zt )

(3.57)

¯ c 0 of the strategy φ ¯ is a supermartingale The tracking error Rφ (X) := π·u (X) − π0u (X) − φ¯ · W ¯ ¯ ¯ 0 ¯ in P ngd (P θ ) given by dQ/dP ¯ under any Q in P ngd (P θ ), and is a martingale under Q 0 = E(λ·W ) with   ¯ t := ht (Zt − φ¯t )A−1 (Zt − φ¯t )−1/2 A−1 Zt − φ¯t + θ¯t , t ∈ [0, T ]. λ t t

Proof. Pointwise existence and uniqueness of θ¯ ∈ Θ follow by the continuity and strict concavity of f θ as a function of θ ∈ Rn , and the uniform boundedness of Θ. Predictability of θ¯ follows by [Roc76]. The claims (3.56) and (3.57) are corollaries of Proposition 3.27. The remaining claims are similar to those of Theorem 3.17, hence their proof goes likewise, making again use of Lemma 3.35 (instead of [Bec09, Lemma 6.1]) and (3.22) and (3.47). ¯ The process φ¯ := φ¯θ in Theorem 3.28 is the good-deal hedging strategy of X for the worst¯ ¯ case model P θ ∈ R which yields that highest good-deal valuation with π·u (X) = π·u,θ (X). ¯ The tracking error of φ¯ is therefore a supermartingale under any measure in P ngd (P θ ) (cf. ¯ Proposition 3.12), i.e. φ¯ is “at least mean-self-financing” under any measure in P ngd (P θ ). However, it is not clear at this stage whether the supermartingale property of the tracking error of φ¯ holds simultaneously under all measures in P ngd (P θ ) for all models R = {P θ : θ ∈ Θ}. We will show that this is the case, and that φ¯ and its associated valuation bound π·u (X) are indeed robust with respect to uncertainty. The idea is first to find an alternative bound π·u,∗ (·) and an associated strategy φ∗ that satisfy the supermartingale property of the tracking error S simultaneously under all measures in θ∈Θ P ngd (P θ ) and are therefore robust. After this, we show that π·u,∗ (X) coincides with the worst-case bound π·u (X), and that the same holds for ¯ the hedging strategies φ(X) and φ∗ (X) for any contingent claim X. In general the good-deal bound π·u (X) is dominated by π·u,∗ (X), but thanks to a saddle point result (Theorem 3.30) one can actually prove that the two bounds are identical. Exchanging the order between ess sup and ess inf in the expression (3.54) for π·u (X), we define for X ∈ L2 (P0 ) and t ≤ T

πtu,∗ (X)

:= ess inf φ∈Φ

ess sup ρθt θ∈Θ



X−

Z t

T



c0 φtr s dWs .

(3.58)

From this it is clear that in general πtu,∗ (X) ≥ πtu (X), for all X ∈ L2 (P0 ). We will show that in fact the minimax identity holds in the sense that the expressions in (3.54) and (3.58) coincide, and that a saddle point exists, giving equality of π·u (X) and π·u,∗ (X). To this end, we describe π·u,∗ (X) and φ∗ in terms of the standard solution (Y, Z) for the BSDE −dYt = f ∗ (t, Zt )dt − Zttr dWt0 ,

t≤T

and

YT = X,

(3.59)

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 129

where f ∗ (t, Zt ) := ess infφ∈Φ f φ (t, Zt ), with f φ (t, Zt ) := ess supθ∈Θ f φ,θ (t, Zt ) for f φ,θ from (3.52). Indeed tr

f φ (t, Zt ) := −ξt0 φt + ess sup θttr (Zt − φt ) + ht (Zt − φt )tr A−1 t (Zt − φt )

1/2

θt ∈Θt

(3.60)

holds and we can identify the robust good-deal hedging strategy φ∗ by Proposition 3.29. Assume (3.22) and (3.47) hold. For X ∈ L2 (P0 ), let (Y, Z) be the standard solution to the BSDE (3.59). Then there exists a unique φ∗ ∈ Φ satisfying φ∗t = argminφ∈Φ f φ (t, Zt ) for t ∈ [0, T ] such that πtu,∗ (X)



= ess inf ρt X − φ∈Φ

Z t

T

c0 φtr s dWs





= ρt X −

Z t

T



c 0 = Yt , t ∈ [0, T ]. φ∗s tr dW s

(3.61)



c 0 is a Q-supermartingale for all Q ∈ P ngd , Moreover Rφ (X) := π·u,∗ (X) − π0u,∗ (X) − φ∗ · W ∗ ∗ ngd ∗ and a Q -martingale for Q ∈ P with dQ /dP0 = E(λ∗ · W 0 ), where −1/2

λ∗ = h (Z − φ∗ )tr A−1 (Z − φ∗ )

A−1 (Z − φ∗ ) + θ∗ ,

with θt∗ := θt∗ (φ∗ ) = argmaxθ∈Θ θttr (Zt − φ∗t ) such that f ∗ (t, Zt ) = f φ

∗ ,θ ∗

(t, Zt ).

Proof. It is clear that for any φ ∈ Φ, there exists θ∗ (φ) ∈ Θ such that θt∗ (φ)tr (Zt − φt ) = ∗ ess supθt ∈Θt θttr (Zt − φt ) and f φ (t, Zt ) = f φ,θ (φ) (t, Zt ). Consider the convex continuous 1/2

tr

function Rn 3 φ 7→ F (φ) := −ξ 0 φ + ess supθ∈Θ θtr (z − φ) + h (z − φ)tr A−1 (z − φ) , for 0 constant h, φ,z, ξ ,σ and A satisfying the notations of Lemma 3.35 and for a compact set Θ ⊂ Rn containing the origin. The function F is also coercive on Im σ tr , i.e. F (φ) → +∞ √ as |φ| → +∞ for Π⊥ (φ) = 0 because ξ 0 < h α0 and ess supθ∈Θ θtr (z − φ) ≥ 0. Hence existence of φ∗ ∈ Φ follows from [ET99, Chapter 1.2]. Uniqueness of φ∗ follows n II, Proposition o ⊥ from the fact that F is strictly convex over Π (φ) = 0 if Π⊥ (z) 6= 0 and strictly convex at φ = z if Π⊥ (z) = 0 because (3.47) holds. Finally, predictability of φ∗ follows from [Roc76, Theorem 2.K] via Part 1 of Proposition 3.2. c0 From Proposition 3.27, for φ ∈ Φ and θ ∈ Θ, Y φ,θ = ρθ· X − ·T φtr s dWs is the Y -component of the solution to the classical BSDE (3.50). As a consequence for every φ ∈ Φ it holds ∗ ess supθ∈Θ f φ,θ (t, Zt ) = f φ,θ (φ) (t, Zt ) = f φ (t, Zt ), t ∈ [0, T ]. The generators f φ are standard, so that by the comparison theorem for classical BSDEs, (Y φ , Z φ ) with Ytφ := ess supθ∈Θ Ytφ,θ is the standard solution to the BSDEs (under P0 ) with parameters (f φ , X), for φ ∈ Φ. The ∗ ∗ ∗ generator f ∗ is also standard because f ∗ (t, Zt ) = f φ ,θ (φ ) (t, Zt ) = ess infφ∈Φ f φ (t, Zt ). Now the comparison theorem yields (3.61) from (3.58). R





The supermartingale property of Rφ (X) can be proved from (3.61) using arguments in the proof of Proposition 3.12. A BSDE proof for the supermartingale property can also be

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 130

given along the same line as the following for the martingale property. By (3.48) it holds ∗ Q∗ ∈ P ngd (P θ ) ⊂ P ngd since λ∗ ∈ C 0 + θ∗ and θ∗ ∈ Θ. Because π u,∗ (X) is the value process of the BSDE (3.59), then after changing measures from P0 to Q∗ one obtains ∗

tr



−dRtφ (X) = f ∗ (t, Zt ) + ξt0 φ∗t − λ∗t tr (Zt − φ∗t ) dt − (Zt − φ∗t )tr dWtQ . 



(3.62)



Furthermore the finite variation part of (3.62) vanishes since f ∗ (t, Zt ) = f φ ,θ (t, Zt ). Because ∗ Rφ (X) ∈ S 2 (P0 ) and dQ∗ /dP0 ∈ Lp (P0 ) for all p < ∞ (since λ∗ is bounded), then H¨older’s ∗ ∗ inequality implies that Rφ (X) ∈ S 2− (Q∗ ) for  ∈ (0, 1). Thus Rφ (X) is a Q∗ -martingale.

Proposition 3.29 shows that the tracking error of the hedging strategy φ∗ with respect to valuation according to π·u,∗ (X) has the supermartingale property simultaneously under all S measures in P ngd = θ∈Θ P ngd (P θ ). The next theorem shows that a minimax identity holds: the sup-inf representation of π·u (·) in (3.54) is equal to the inf-sup representation of π·u,∗ (·) in (3.58); see also (3.63). Moreover, the good-deal hedging strategy φ¯ with respect to the ¯ that gives the highest good-deal valuation bound π u (·), is worst-case model (given by θ) · identical with the robust good-deal hedging strategy φ∗ from Proposition 3.29. Theorem 3.30. Assume (3.22) and (3.47) hold. For X ∈ L2 (P0 ), let (Y, Z) be standard solution of the BSDE (3.59). Then fφ

∗ ,θ ∗

¯¯

(t, Zt ) = ess inf ess sup f φ,θ (t, Zt ) = ess sup ess inf f φ,θ (t, Zt ) = f φ,θ (t, Zt ) (3.63) φ∈Φ

θ∈Θ

θ∈Θ

φ∈Φ

¯ θ), ¯ (φ∗ , θ∗ ) from Theorem 3.28 and Proposition 3.29. Moreover (Y, Z) coincides holds with (φ, with the standard solution to the BSDE (3.55) and πtu (X) = πtu,∗ (X) = Yt

and φ∗t (X) = φ¯t (X),

t ∈ [0, T ].

(3.64)

Proof. Let X ∈ L2 (P0 ). By an application of Lemma 3.36, the generator f φ,θ of the BSDE (3.50) for θ ∈ Θ and φ ∈ Φ satisfy the minimax relation (3.63). By Theorem 3.28 and ∗ ∗ ¯¯ Proposition 3.29 it holds f (t, Zt ) = f φ,θ (t, Zt ) and f ∗ (t, Zt ) = f φ ,θ (t, Zt ), t ∈ [0, T ], for f, f ∗ respectively generators of the BSDEs (3.55), (3.59). Also, πtu (X) = πtu,∗ (X) = Yt , t ∈ [0, T ], since by uniqueness of BSDE solutions (Y, Z) also solves the BSDE (3.55). Hence ¯ θ) ¯ and (φ∗ , θ∗ ) are both saddle points of the function (φt , θt ) 7→ f φ,θ (t, Zt ). Now for any (φ, 1/2 tr θ ∈ Θ and z ∈ Rn , the function φ 7→ F (φ, θ) := θtr (z − φ) − ξ 0 φ + h (z − φ)tr A−1 (z − φ) is strictly over {Π⊥ (φ) = 0} if Π⊥ (z) 6= 0, and strictly convex at φ = z if Π⊥ (z) = 0, convex √ θ since ξ < h α0 . [ET99, Chapter VI, Proposition 1.5] implies that the φ-components of the saddle points are identical, yielding φ¯ = φ∗ .

Section 3.3. Good-deal valuation and hedging under model uncertainty

3.3.5

Page 131

The impact of model uncertainty on robust good-deal hedging

In the framework of Section 3.3.4, results have so far been stated for an arbitrary standard correspondence Θ without further structural assumptions, and ellipsoidal correspondences were only assumed for the no-good-deal constraints C θ , θ ∈ Θ. Recall (cf. Theorem 3.17 and subsequent remarks) that in the absence of uncertainty the good-deal hedging strategy contains a speculative component in the direction of the market price of risk. This already indicates that under uncertainty one should expect to see relevant differences by a robust approach to hedging. To investigate the effect of uncertainty about the market price of risk θ on robust good-deal hedging, we assume in addition (noting that θ ∈ Im σ tr is natural) that for all (t, ω) ∈ [0, T ] × Ω, the set Θt (ω) is a subset of Im σttr (ω) in the sense that (3.65)

Θt (ω) = Θ0t (ω) ∩ Im σttr (ω)

holds for some standard correspondence Θ0 with 0 ∈ Θ0 satisfying the uniform boundedness Assumption 3.3. With (3.65), one clearly has Π⊥ (θ) = 0 for all θ ∈ Θ. This leads to the following simplified expressions of the BSDE generators f φ,θ , f θ : tr

1/2

, and

1/2

.

f φ,θ (t, z) = θttr (Πt (z) − φt ) − ξt0 φt + ht (z − φt )tr A−1 t (z − φt ) f θ (t, z) =

tr −ξt0 Πt (z)

+ h2t −

1/2 tr ξtθ At ξtθ

tr

−1 ⊥ Π⊥ t (z) At Πt (z)

¯ As a consequence, the process θ¯ = θ(X) does actually not depend on the contingent claim 2 X ∈ L (P0 ) under consideration, and solves the minimization problem ¯ tr

tr

¯

ξtθ At ξtθ = min ξtθ At ξtθ , θt ∈Θt ¯

t ∈ [0, T ].

(3.66)

In addition in this case, one has Qngd (P θ ) = θ∈Θ Qngd (P θ ) = Qngd . To obtain even more explicit results one may assume e.g. ellipsoidal uncertainty n

S

o

for all (t, ω) ∈ [0, T ] × Ω,

Θ0t (ω) := x ∈ Rn | xtr Bt (ω)x ≤ δt2 (ω)

(3.67)

with δ being a positive bounded and predictable process, and B being a uniformly elliptic and predictable matrix-valued process, satisfying the separability condition (3.22) with respect to σ. Clearly f φ (t, Zt ) from (3.60) in this case is equal to tr

1/2

−ξt0 φt + δt (Πt (Zt ) − φt )tr Bt−1 (Πt (Zt ) − φt )

+ ht (Zt − φt )tr A−1 t (Zt − φt )

1/2

.

In terms of φ∗ and the solution (Y, Z) to the BSDE (3.59), the process θ∗ = θ∗ (φ∗ ) of Proposition 3.29 is given by θt∗ (X) = δt (Πt (Zt ) − φ∗t )tr Bt−1 (Πt (Zt ) − φ∗t )

−1/2

Bt−1 (Πt (Zt ) − φ∗t ).

(3.68)

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 132

Remark 3.31. Let us recall Remark 3.24 b). In the present context of Section 3.3.5 with constraints of ellipsoidal type for good-deals (3.46) and for model uncertainty (3.67), results as explicit as in Section 3.2.1 can be obtained in particular cases, as elaborated subsequently, but not in general. Indeed, using ξ θ = ξ 0 + θ for θ ∈ Θ, to find the minimizer θ¯ (the worst-case) in (3.66) requires to compute the projection of −ξt0 onto the ellipsoid Θt with respect to the norm induced by the matrix At . In the radial case A ≡ IdRn the projection is Euclidian. While there is no closed formula for the projection in general, the solution is described by a parametric formula in terms of a Lagrangian multiplier that solves a 1-dimensional equation, and it can be computed by efficient algorithms (see [Kis94]) even if this operation may be required frequently (as in Monte Carlo simulation, cf. Section 3.2.2). It is instructive to look at the special case where in addition the matrices A and B are related through B = A/r for some scalar r > 0; in other words, B basically equals A up to a change √ of δ to rδ. In this case (3.66) is solved by √ rδt 0 ¯ (3.69) θt = −ξt I{ξ0 tr At ξ0 ≤rδ2 } − ξt0 I{ξ0 tr At ξ0 >rδ2 } , t ∈ [0, T ], tr 0 t t t t t t (ξt At ξt0 )1/2 and replacing φ∗ = φ¯ in the formula of θ∗ in (3.68) by its expression from (3.57) in terms of θ¯ ¯ Note that (3.69) implies that θ¯tr Aθ¯ is equal to ξ 0 tr Aξ 0 on {ξ 0 tr Aξ 0 ≤ rδ 2 } one obtains θ∗ = θ. tr and equal to rδ 2 on {ξ 0 Aξ 0 > rδ 2 }. In other words, the worst-case Girsanov kernel −θ¯ is equal to the market price of risk ξ 0 of the center P0 of the confidence set R of reference measures, being truncated such that θ¯tr Aθ¯ = rδ 2 holds for large values of ξ 0 outside of the ellipsoidal set {x ∈ Rn : xtr Ax ≤ rδ 2 }. To obtain an intuition about the impact that model uncertainty may have on robust good-deal hedging, let us look at the behavior of the worst-case Girsanov kernel θ¯ = θ∗ obtained in (3.69) and the hedging strategy φ¯ = φ∗ in (3.57) for varying scaling constant r: As r becomes large, the worst-case Girsanov kernel −θ¯ becomes close to the market price of risk ξ 0 and φ∗ = φ¯ close to Π(Z). This shows that as uncertainty becomes overwhelming, the robust good-deal hedging strategy ceases to comprise a speculative component in the direction of the market price of risk. In such a situation one can show that the hedging strategy is the risk-minimizing ¯ strategy under the worst-case no-good-deal measure in the worst-case model P θ . More precisely, for an arbitrary shape of the correspondence Θ0 , if uncertainty is big enough for the confidence set R of reference measures to contain some risk neutral pricing measure from Me , then robust good-deal hedging for any claim X does not comprise a speculative component and the holdings φ∗ of the hedging strategy in risky assets coincide with those of the globally risk-minimizing strategy by [FS86] (cf. also [Sch01, Section 2]) under worst-case no-good-deal ¯ ¯ ¯ ¯ ¯ = Q(X, ¯ measure Q P θ ) ∈ Qngd (P θ ), i.e. satisfying πtu (X) = πtu,θ (X) = EtQ [X] for any ¯ t ∈ [0, T ], for the worst-case model P θ . Note that here risk-minimization is under a risk-neutral ¯ that could also depend on the contingent claim into consideration, and not under measure Q

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 133

b as in the original works [FS86, Sch01]. The eventually the minimal martingale measure Q non-speculative nature of the robust good-deal hedging strategy under (large) uncertainty offers new theoretical support for the quadratic hedging objective of risk minimization, which may be criticized for giving equal weighting for upside and downside risk. More broadly, such gives support to a common perception (see e.g. [LP00]) that speculative objectives should be avoided in hedging, in addition to more practical arguments like simplifications for marking-to-market (uses risk neutral valuation). To make the said statement precise, consider the classical BSDE tr

tr

1/2 

−1 ⊥ −dYt = − ξt0 Πt (Zt ) + ht Π⊥ t (Zt ) At Πt (Zt )

dt − Zttr dWt0 ,

(3.70)

for t ∈ [0, T ] with YT = X. First we prove the following Proposition 3.32. Assume (3.22) and (3.47) hold and that Θ satisfies (3.65). For any X ∈ L2 (P0 ), let (Y X , Z X ) denote the standard solution of the BSDE (3.70). Then π·u (X) = Y X and φ∗ (X) = Π(Z X ) for all X ∈ L2 (P0 ) e

R ∩ M (S) 6= ∅.

holds, if and only if

(3.71) (3.72)

Proof. Let X ∈ L2 (P0 ). Recall that for Θ defined in (3.65), θ¯ from Theorem 3.28 does not vary with X and solves the minimization problem (3.66). Now if (3.72) holds, then there exists b θ = P θ , and therefore ξ θ = 0. This implies that θ ∈ Θ such that P θ ∈ R ∩ Me (S) 6= ∅, i.e. Q ¯ 0 0 θ = θ¯ = −ξ and hence ξ ∈ Θ. As a consequence, the generator f = f θ of the BSDE (3.55) coincides with that of the BSDE (3.70). By uniqueness of standard BSDE solutions follows π·u (X) = Y X . Now from Theorem 3.28 and Theorem 3.30 one obtains that φ∗ = φ¯ = Π(Z X ). ¯ Conversely, suppose that (3.71) holds. Then the generator f = f θ for the BSDE (3.55) and the one for (3.70) are equal everywhere by [CHMP02, Theorem 7.1 and Rmk. 4.1]. This implies ¯ b θ¯ = P θ¯, and hence R ∩ Me (S) 6= ∅. (since Π⊥ (θ) = 0 for all θ ∈ Θ) that ξ θ = 0, i.e. Q

Now we can make the previously described relation between global risk minimization and good-deal hedging under (large) uncertainty precise. Theorem 3.33. Let the assumptions of Proposition 3.32 and (3.72) hold. For X in L2 (P0 ), let (Y, Z) be the standard solution of the BSDE (3.70). Then π·u (X) = Y has the GKW c 0 (and S = diag(S)σ · W c 0 , cf. Section 3.1.1) decomposition with respect to σ · W ∗

c 0 + Rφ , πtu (X) = π0u (X) + φ∗ · W t t

t ∈ [0, T ], ¯



(3.73)

Q ¯ with φ∗ = Π(Z). The tracking error Rφ (X) = Π⊥ · (Z) · W  is a Q-martingale orthogonal to 0 c , for Q ¯ ∈ Qngd given by dQ/dP ¯ σ·W ¯) · W 0 with 0 = E (−ξ + η tr

−1 ⊥ η¯t = ht Π⊥ t (Zt ) At Πt (Zt )

−1/2

⊥ A−1 t Πt (Zt ),

t ∈ [0, T ].

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 134

Proof. From Proposition 3.32 we have φ∗ = Π(Z) and π·u (X) = Y . By the definitions of η¯ ¯ Yt = Y0 + Z · W Q¯ holds for all t ∈ [0, T ]. As a consequence, one obtains πtu (X) = and Q, t c 0 +Π⊥ (Z· )·W Q¯ , t ∈ [0, T ]. Thus (3.73) holds with Rφ∗ (X) = Π⊥ (Z)·W Q¯ being π0u (X)+φ∗ · W t · · t R ¯ ¯ a Q-martingale orthogonal to S = S0 + 0· diag(St )σt dWtQ since σ(Π⊥ (Z)) = 0. Furthermore · ¯ ⊥ ∗ ∗ 0 ∗ Q φ∗ c c 0 under since φt ⊥Πt (Zt ), and φ · W = φ · W , then R (X) is also orthogonal to φ∗ · W ¯ Therefore (3.73) is the GKW decomposition of π ¯ Q. ¯·u (X) under Q. c 0 , any Galtchouk-KunitaRemark 3.34. Note that using Section 3.1.1 and dS/S = σdW Watanabe (GKW) decomposition (see [Sch01]) of a continuous local Q-martingale M for Q in c 0 gives a GKW decomposition with respect to S and vice versa. In Qngd with respect to σ · W this sense, Theorem 3.33 shows that the robust good-deal hedging strategy φ∗ for X coincides with the (global) risk-minimizing strategy of [FS86] (cf. [Sch01, Section 2]) with respect to a b 0 ) under which π u (X) is equal to E Q¯ [X] ¯ = Q(X) ¯ specific measure Q ∈ Qngd (instead of Q t t b0 Q 0 b ¯ (instead of E [X]), t ∈ [0, T ]. Note that Q(X) is not equal to Q in general unless h = 0, in t

which case equality holds for any contingent claim X ∈ L2 (P0 ), or X is replicable. A seminal no-trade result by [DW92] shows that a utility optimizing agent abstains from taking any position in a tradeable risky asset if uncertainty is too large. In comparison, the above theorem shows that a good-deal hedger keeps dynamically trading according to the risk minimizing component Π(Z) but ceases to comprise any speculative component. [BCCH14] demonstrate by numerical computation in an example, in a setting quite different to ours, that the relative benefit of dynamic hedging compared to static hedging could decrease if uncertainty increases. This is intuitive, since (see e.g. [Con06]) static hedges can be less exposed to model risk. Proposition 3.32 likewise addresses how increasing uncertainty affects dynamic hedging, but is different in that it offers theoretical conditions under which dynamic good-deal hedging φ∗ ceases to comprise speculative components in order to compensate for exposures to non-spanned risk.

3.3.6

Example with closed-form solutions under model uncertainty

The usual filtration is generated by a two-dimensional Brownian motion W 0 = (W 0,S , W 0,H )tr under P0 . We consider a single traded risky asset with price S and a non-traded asset with value H modelled under P0 for t ∈ [0, T ] by dSt = St σ

S

ξ

0,S

dt +

 dWt0,S ,

dHt = Ht γdt +

β(ρdWt0,S

+

q

1 − ρ2 dWt0,H )



with S0 , H0 > 0, scalars σ S , β > 0, γ, ξ 0,S ∈ R and correlation coefficient ρ ∈ [−1, 1]. We derive robust good-deal bounds and hedging strategies in closed-form, for European call options on the non-traded asset and for no-good-deal constraint and uncertainty modelled (as in

Section 3.3. Good-deal valuation and hedging under model uncertainty

Page 135

Section 3.3.5) using the radial sets C 0 = {x ∈ R2 : |x| ≤ h} and Θ0 = {x ∈ R2 : |x| ≤ δ} for scalars h, δ ≥ 0. Here one has Θ = Θ0 ∩ Im σ = [−δ, δ] × {0}, for σ = (σ S , 0), and hence ξ θ = (ξ θ,S , 0)tr := (ξ 0,S + θS , 0)tr ∈ Im σ, for models P θ with θ = (θS , 0)tr , where θS ∈ [−δ, δ]. ¯ From (3.69), with A = B ≡ IdR2 and r = 1, the worst-case model P θ corresponds to ξ 0,S θ¯S = −ξ 0,S I{|ξ0,S |≤δ} − δ 0,S I{|ξ0,S |>δ} . |ξ |

(3.74)

By Theorems 3.28,3.30, the robust good-deal bound and hedging strategy for a call option  tr 2 ¯ θ,S ¯ , X := (HT − K)+ are given by π·u (X) = Y and φ(X) = Z 1 + √ 2|Z |θ,S ξ 0 , for ¯ 2 h −|ξ

|

¯ (Y, Z := (Z 1 , Z 2 )tr ) solving the BSDE (3.55), equaling the BSDE (3.51) for θ = θ: ¯

¯

¯

−dYt = − ξ θ,S Zt1 + (h2 − |ξ θ,S |2 )1/2 |Zt2 | dt − Zttr dWtθ 

and YT = X,

(3.75)

¯ b θ¯ and using (3.74), with Wtθ := (Wt0,S − θ¯S t, Wt0,H )tr , t ∈ [0, T ]. Writing (3.75) under Q arguments analogous to those in the derivation of (3.31) yield

πtu (X) = N (d+ )Ht eα˜ + (T −t) − KN (d− ) =: eα˜ + (T −t) ∗ B/S-call-price time: t, spot: Ht , strike: Ke−α˜ + (T −t) , vol: β , 

πtl (X) = eα˜ − (T −t) ∗ B/S-call-price time: t, spot: Ht , strike: Ke−α˜ − (T −t) , vol: β , 

with

√    1  d± := ln Ht /K + α ˜ + ± β 2 (T − t) / β T − t , 2 q

˜ 1 − ρ2 α ˜ ± := γ + β − ρξ 0,S ± h



and 1/2 ¯ 2 1/2 ˜ = h(δ) ˜ h := h2 − |ξ θ,S |2 = hI{|ξ0,S |≤δ} + h2 − ξ 0,S − δ I{|ξ0,S |>δ}



. Analogously to the derivation of (3.32), note that Z = eα˜ + (T −t) N (d+ )Ht β(ρ, Hence the (seller’s) robust good-deal hedging strategy is obtained as α ˜ + (T −t)

φ¯t (X) = e



N (d+ )Ht β ρ +

p

1 − ρ2 ξ 0,S e 0,S | h|ξ

tr

|ξ 0,S | − δ 1{|ξ0,S |>δ} , 0 

1 − ρ2 )tr .

p

, t ∈ [0, T ].

¯ For |ξ 0,S | > δ, the speculative nature of φ(X) is reflected by the presence of the second ¯ summand in the first component of φ(X) above. For |ξ 0,S | ≤ δ, this summand vanishes and the function δ 7→ α ˜ + is constant on δ ∈ [|ξ 0,S |, ∞]. In this case robust good-deal hedging is ¯ = Qλ¯ ∈ Qngd (P0 ) with Girsanov then globally risk-minimizing with respect to the measure Q ¯ := − ξ 0,S , htr and non-speculative as proved in Theorem 3.33. Note that for kernel λ b 0 in absence of uncertainty), we recover δ = ξ 0,S = 0 (i.e. risk-neutral setting under P0 = Q formulas of Section 3.2.2 for n = 2 and d = 1.

Section 3.4. Appendix

Page 136

Figure 3.4 illustrates the dependence of the bounds π0u (X) and π0l (X) in the presence of uncertainty, on the correlation coefficient ρ, uncertainty size δ and no-good-deal constraint (optimal growth rate bound) h, and for global parameters γ = 0.05, β = 0.5, K = 1, H0 = 1 and T = 1. Figures 3.4a,3.4b are plots of π0u (X) and π0l (X) as functions of ρ for fixed δ = 0 (i.e. absence of uncertainty) and ξ 0,S ∈ {0, 0.2}, showing how the good-deal bounds vary for different values of h. Figure 3.4d contains a similar plot for fixed h = ξ 0,S = 0.2, showing how the bounds vary with ρ for different values of δ. One can observe that the maximum of π0u (X) and minimum of π0l (X) are attained at ρ = 0 only for ξ 0,S = 0 (cf. Figure 3.4a). b 0 ), then the largest In other words, if the market price of risk ξ 0,S is zero (hence P0 = Q good-deal bounds are obtained when the traded and non-traded assets are uncorrelated (i.e. ρ = 0). On the other hand if ξ 0,S > 0 (as e.g. in Figures 3.4b,3.4d), the plots are tilted so that the maximum of π0u (X) (resp. minimum of π0l (X)) is reached at ρ < 0 (resp. ρ > 0). For π0u (X), this is explained by the fact that if the market price of risk ξ 0,S is positive, the ¯ = Qλ¯ ∈ Qngd (P0 ) with supremum in (3.3) is maximized by the no-good-deal measure Q  ¯ := − ξ 0,S , h ˜ tr under which the upward drift α Girsanov kernel λ ˜ + of the underlying price process H is maximized, clearly at a negative correlation ρ. The explanation for π0u (X) is similar, with α ˜ − being minimal at a positive correlation, for ξ 0,S > 0. For ξ 0,S < 0 the tilt of the plots occurs in the other direction. That the good-deal bounds in Figures 3.4a,3.4b,3.4d coincide for perfect correlation ρ = ±1 is clear, because in this case derivatives X on H are attainable and admit unique no-arbitrage prices, implying π·u (X) = π·l (X). Finally, Figure 3.4c illustrates the evolution with respect to δ of the good-deal bounds at time t = 0 for ρ = 0.6, ξ 0,S = 0.2 and different values of h, with |ξ 0,S | chosen as the smallest value h0 of h. One observes that for each given h, the good-deal bound curves become flat for δ ≥ |ξ 0,S | (as predicted by Proposition 3.32), and match (i.e. π0u (X) = π0l (X)) for δ = |ξ 0,S | − h0 = 0 (as might be expected in the absence of uncertainty for a degenerate expected growth rate bound h = |ξ 0,S |).

3.4

Appendix

This appendix includes lemmas and proofs omitted from the main body of the chapter. For the convenience of the reader, some derivations are detailed as well. Lemma 3.35. For d < n, let σ ∈ Rd×n be of full-rank, A ∈ Rn×n be symmetric and positive definite, and h > 0, Z ∈ Rn , ξ ∈ Im σ tr . Let α0 > 0 be a constant of ellipticity √ of A−1 and assume that |ξ| < h α0 and A−1 (Ker σ) = Ker σ. Then the vector φ¯ := 1/2 2 −1/2 Π(Z) + Π⊥ (Z)tr A−1 Π⊥ (Z) h − ξ tr Aξ Aξ is the unique minimizer of the function φ 7→ F (φ) := −ξ ∗ φ + h (Z − φ)tr A−1 (Z − φ)

1/2

on Im σ tr .

Proof. Since A−1 (Ker σ) = Ker σ, then φ¯ ∈ Im σ tr . The Kuhn-Tucker optimality conditions

Section 3.4. Appendix

(a)

h0 = ξ0,S = 0, δ = 0

(c)

h0 = ξ0,S = 0.2, ρ = 0.6

Page 137

(b)

h0 = ξ0,S = 0.2, δ = 0

(d)

h = ξ0,S = 0.2

Figure 3.4: Dependence of π0u (X), π0l (X) on ρ, h and/or δ ¯ The function F is convex and differentiable at every φ 6= Z, where its are satisfied by φ. −1/2 −1 ¯ = gradient is ∂F (φ) = −ξ − h (Z − φ)tr A−1 (Z − φ) A (Z − φ). This yields ∂F (φ)   1/2 −1/2 6 Z, using A−1 (Ker σ) = Ker σ. − h2 − ξ tr Aξ Π⊥ (Z)tr A−1 Π⊥ (Z) A−1 Π⊥ (Z) for φ¯ = √  At φ = Z the subgradient is well-defined and the inclusion E := x ∈ Rn |x| ≤ h α0 −|ξ| ⊆   ¯ = − h2 − ξ tr Aξ 1/2 Π⊥ (Z)tr A−1 Π⊥ (Z)−1/2 A−1 Π⊥ (Z) for ∂F (φ) holds. Overall ∂F (φ) ¯ for φ¯ = Z. In any case, φ¯ satisfies the Karush-Kuhn-Tucker φ¯ 6= Z and 0 ∈ E ⊆ ∂F (φ) conditions and since F is convex and the minimization constraint φ ∈ Im σ tr is linear, optimality of φ¯ follows from the Kuhn-Tucker theorem (cf. [Roc70, Section 28]). Uniqueness of φ¯ is implied by the fact that F is strictly convex over Im σ tr if Π⊥ (Z) 6= 0 and strictly convex at φ¯ √ if Π⊥ (Z) = 0 since |ξ| < h α0 .

Lemma 3.36. Let d < n, h > 0 be constant, Z ∈ Rn , A ∈ Rn×n a symmetric positive definite matrix, σ ∈ Rd×n a full (d)-rank matrix, and ξ 0 ∈ Φ := Im σ tr . Let Θ ⊂ Rn be a convex1/2 tr compact set, and F : Rn × Rn 3 (φ, θ) 7→ θtr (Z − φ) − ξ 0 φ + h (Z − φ)tr A−1 (Z − φ) . Then the minmax identity inf φ∈Φ supθ∈Θ F (φ, θ) = supθ∈Θ inf φ∈Φ F (φ, θ). holds. Proof. For all φ ∈ Rn , the function θ 7→ F (φ, θ) is concave, continuous. For all θ ∈ Rn the

Section 3.4. Appendix

Page 138

function φ 7→ F (φ, θ) is convex and continuous. As Θ ⊂ Rn is convex and compact, and Φ = Im σ tr is convex and closed, a minimax theorem [ET99, Chapter VI, Proposition 2.3] applies and the minmax identity holds.

Proof of Lemma 3.1. Part a) is classical (see [Del06] and cf. previously given other references). As for Part b), m-stability and convexity of Me follow from [Del06, Proposition 5]. Convexity of Qngd follows from that of Me and the values of C. To show m-stability of Qngd , let Z i = E(λi · W ) ∈ Qngd , i = 1, 2, τ ≤ T be a stopping time and Z = I[0,τ ] Z·1 + I]τ,T ] Zτ1 Z·2 /Zτ2 . Since Me is m-stable, then Z ∈ Me and one has Z = E(λ · W ) for some predictable process λ. It remains to show that λ is bounded and that λ ∈ C. From the expression of Z, writing the densities Z, Z 1 , Z 2 as ordinary exponentials by distinguishing t ≤ τ and t ≥ τ , and taking the 2 2  R logarithm yields (λ − I[0,τ ] λ1 − I]τ,T ] λ2 ) · W = 12 0· |λs |2 − I[0,τ ] (s) λ1s − I]τ,T ] (s) λ2s ds. Since F is the augmented Brownian filtration, then [0,   τ ] and ]τ, T ] are predictable and so is 1 2 1 2 λ − I[0,τ ] λ − I]τ,T ] λ . Hence λ − I[0,τ ] λ − I]τ,T ] λ · W is a continuous local martingale of finite variation and is thus equal to zero. As a consequence λ = IB λ1 + IB c λ2 is bounded since λ1 , λ2 are, and satisfies λ ∈ C since C is convex-valued.

Proof of Theorem 3.7. Without loss of generality, we argue only for X ≥ 0; otherwise one can use translation invariance with X + kXk∞ ≥ 0. et (X) := ess sup Part 1: Let t ∈ [0, T ] and define π E Q [X]. We have the inclusions Q∈Qngd t

Ctk (ω) ⊆ Ctk+1 (ω) ⊆ Ct (ω) for all (t, ω) and for all k ∈ N, and hence the chain of inet (X) holds for k ∈ N. Since the sequence equalities πtu,k (X) ≤ πtu,k+1 (X) ≤ πtu (X) ≤ π u,k u,k (πt (X))k∈N is non-decreasing and |πt (X)| ≤ kXk∞ , for all k, the monotone a.s. limit et (X) ≥ Jt . It remains to show the reverse inequality, Jt := limk%∞ πtu,k (X) is finite and π et (X) = πtu (X), using Part 2 of Proposition 3.5 to π·u,k (X) to obtain a sequence which implies π ¯ k ∈ Qngd ⊆ Qngd satisfying π u (X) ≥ E Q¯ k [X] = π u,k (X) % π et (X) as k → ∞. of measures Q t t t k To this end, it suffices to show that J is a c`adl`ag Q-supermartingale for all Q ∈ Qngd and ngd

e· (X) = π·u,Q (X) since Qngd is also convex and m-stable (arthen apply Lemma 3.6 to π gument being analogous to that for Qngd in Lemma 3.1) b). First notice that J is a c`adl`ag Q-supermartingale for any Q ∈ Qngd with Girsanov kernel λQ = λ. Indeed for such measures Q, there exists k0 ∈ N such that λ ∈ Λk for all k ≥ k0 . Since Jt = limk πtu,k (X) and π·u,k (X) is a bounded c`adl`ag Q-supermartingale for every k ≥ k0 , then J is a c`adl`ag Q-supermartingale as the increasing limit of c`adl`ag Q-supermartingales of class D (cf. [Doo01, Section 2.IV.4]). Now let Q ∈ Qngd with λQ = λ = −ξ + η ∈ Λ not necessarily bounded. Then λn := −ξ + η n with η n := ηI{|η|≤n} ∈ Ker σ forms a sequence of bounded Girsanov kernels for measures

Section 3.4. Appendix

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Qn ∈ Qngd such that limn→∞ λn = λ P ⊗ dt − a.e.. By the above arguments, since ξ and X b are bounded, then J is a bounded c`adl`ag Q-supermartingale (hence of class D) that admits a c with respect Doob-Meyer decomposition which, by the predictable representation property of W b F) (cf. [HWY92, Theorem 13.22]), reads J = J0 + Z · W c − A, where Z ∈ H2 (Q) b and A (Q, 2 b is a non-decreasing predictable processes with A0 = 0 and AT ∈ L (Q) because J ∈ S ∞ is bounded (cf. [DM82, Inequality (15.1), Section VII.15, page 202]). One rewrites n

J = J0 + Z · W Q + J = J0 + Z · W Q +

·

Z

Z 0

0 ·

Zttr ηtn dt − A, and

Zttr ηt dt − A.

(3.76) (3.77)

Since the Girsanov kernels λn are bounded, J is a c`adl`ag supermartingale under Qn for all n. Hence from (3.76) one has dAt ≥ Zttr ηtn dt, t ∈ [0, T ], for all n ∈ N. By dominated convergence, taking the limit as n → ∞ implies dAt ≥ Zttr ηt dt, t ∈ [0, T ]. Now since X is non-negative, R then so is J and positivity of − 0· Zttr ηt dt + A implies Z · W Q ≥ const from (3.77). Being bounded from below, the local Q-martingale Z · W Q is therefore a Q-supermartingale. Finally R because J is bounded, then 0T Zttr ηt dt − AT is Q-integrable and thus J is a Q-supermartingale. Part 2: For k ∈ N, the process π u,k is the good-deal bound associated to the constraint correspondence C k satisfying Assumption 3.3. Hence applying Part 2 of Proposition 3.5 with C replaced by C k yields the result. b ∈ Qngd ⊂ Qngd . Hence by Lemma 3.6, π u (X) and π·u,k (X) Part 3: For all k ≥ kξk∞ holds Q · k b b as are bounded c`adl`ag Q-supermartingales, admitting Doob-Meyer decompositions under Q k in (3.12), and by Part 2 A satisfies (3.13). By arguments similar to those for Part 1 follows b and A, Ak ∈ L2 (Q). b Z, Z k ∈ H2 (Q) b Fu ) for all u ≤ T follows Part 4: From Part 3, that Aku converges to Au weakly in L2 (Ω, Q, from [DM82, Theorem VII.18 and subsequent remarks]. These apply since the sequence (π·u,k (X))k≥kξk∞ is uniformly bounded by kXk∞ , and hence Part 1 and dominated convergence b Fu ), for all u ∈ [0, T ]. Furthermore the imply that πuu,k (X) converges to πuu (X) in L2 (Ω, Q, u,k k cu → Z · W cu weakly in L2 (Ω, Q, b Fu ) convergences of (πu (X))k and (Au )k imply that Z k · W for all u ∈ [0, T ]. By the predictable representation property and Itˆo’s isometry, follows b ⊗ dt) for any u. Z k → Z weakly in L2 (Ω × [0, u], Q

b the Doob-Meyer decomposition Proof of Theorem 3.9. By Theorem 3.7, π·u (X) admits under Q R · c − A = π u (X) + Z · W + ξ tr Zt dt − A, where Z ∈ H2 (Q) b π·u (X) = π0u (X) + Z · W 0 0 t and A is a non-decreasing predictable process with A0 = 0. Alternatively one rewrites R R −dπtu (X) = gt (Zt )dt − Zttr dWt + dKt , with K := A − 0· ξttr Zt dt − 0· ess supλt ∈Λt λtr t Zt dt u being finite-valued and predictable. For (π· (X), Z, K) to be a supersolution to the BSDE with

Section 3.4. Appendix

Page 140

parameters (g, X) it suffices to show that K is non-decreasing. For any λ = −ξ + η ∈ Λ, one can construct the sequence of λn = −ξ + η n ∈ Λ Girsanov kernels of measures Qn ∈ Qngd with η n = ηI{|η|≤n} such that λn → λ P ⊗ dt-a.s. as n → ∞. For each Qn it holds R n π·u (X) = π0u (X) + Z · W Q + 0· Zt tr ηtn dt − A. Since π·u (X) is a bounded Qn -supermartingale, then dAt − ξttr Zt dt ≥ Zttr λnt dt, for all n ∈ N. Taking the limit as n → ∞ and using dominated convergence one obtains dAt − ξttr Zt dt ≥ Zttr λt dt. Now taking the essential supremum over all λ ∈ Λ yields dKt ≥ 0. To show that the supersolution (π·u (X), Z, K) is minimal, it suffices (by Lemma 3.6) to show that the Y -component of any other supersolution is a c`adl`ag Q-supermartingale for every ¯ K) ¯ be a supersolution of the BSDE with parameters (g, X), with Q ∈ Qngd . Let (Y¯ , Z, ∞ ¯ Y ∈ S . By change of measure, the dynamics of Y¯ under some measure Q ∈ Qngd with Girsanov kernel λQ ∈ Λ is 



Q ¯ ¯ tr Q ¯ tr ¯ −dY¯t = ess sup λtr t Zt − Zt λt dt − Zt dWt + dKt ,

t ∈ [0, T ].

λt ∈Λt

(3.78)

¯ is non-decreasing, it holds that Since K 



¯ t + ess sup λtr ¯ ¯ tr Q dK t Zt − Zt λt dt ≥ 0,

t ∈ [0, T ].

λt ∈Λt

(3.79)

From (3.79), (3.78) and boundedness of Y¯ , the local martingale Z¯ · W Q is bounded from below, hence is a supermartingale. Again since Y¯ ∈ S ∞ , then the integral of (3.79) in [0, T ] is Q-integrable and therefore Y¯ is a Q-supermartingale.

Proof of Corollary 3.10. By m-stability and convexity of Qngd , Lemma 3.6 and Part 1. of ¯ Theorem 3.7 imply that π·u (X) is a c`adl`ag Q-supermartingale with terminal value X since ngd ¯ ∈ Q . We have Q ¯

¯

¯

¯

¯

E Q [X] = π0u (X) ≥ E Q [πtu (X)] ≥ E Q [EtQ [X]] = E Q [X], t ≤ T. b from Theorem 3.9 with ¯ Hence π·u (X) is a Q-martingale. Let Z ∈ H2 (Q)

K := A −

Z 0

·

ξttr Zt dt −

Z 0

·

ess sup λtr t Zt dt λt ∈Λt

such that (π·u (X), Z, K) is the minimal supersolution to the BSDE with parameters (g, X). R ¯ t + ξt )dt − A. Since ¯ as π·u (X) = π u (X) + Z · W Q¯ + · Zttr (λ One writes π·u (X) under Q 0R 0 · ¯ t + ξt )dt = 0. Therefore since λ ¯ ∈ Λ, one ¯ π·u (X) is a bounded Q-martingale, then A − 0 Zttr (λ R · tr ¯ t + ξt )dt = 0. Thus K = 0 and hence (π u (X), Z) is a BSDE obtains 0 ≤ K ≤ A − 0 Zt (λ · solution. Any solution being a supersolution, minimality follows from Theorem 3.9. Finally

Section 3.4. Appendix

Page 141

¯ tr Zt dt − Z tr dWt , with π u (X) = X, then ¯ since the Q-martingale π·u (X) satisfies −dπtu (X) = λ t t T ¯ tr Zt holds. ess supλt ∈Λt λtr Z = λ t t t

˜ T , S˜T ) ∈ L2 for a payoff function Derivation of (3.33). Consider a European option X = G(H (0, ∞)2 3 (x, y) 7→ G(x, y) ∈ R being measurable, non-decreasing in x and at most of polynomial growth in x±1 , i.e. |G(x, y)| ≤ k(1 + xn + x−n ) for all (x, y) ∈ (0, ∞)2 , for some k > 0 and n ∈ N. Again following the arguments of the proof in the example of an option on ˜ one can show that H, ¯=h λ

n X

−1/2 β˜i2 /ai (0, . . . , 0, β˜d+1 /ad+1 , . . . , β˜n /an )tr

i=d+1 ¯ ˜ t , S˜t ) for u ∈ C (0, ∞) × (0, ∞)2 with ∂x u ≥ 0. ˜ T , S˜T )] = u(t, H and πtu (X) = EtQ [G(H Moreover one obtains for all t ∈ [0, T ] that



 Z i = t Z i = t

˜ t ∂x u(t, H ˜ t , S˜t ) + σ ˜ t , S˜t ), for i ≤ d and β˜i H ˜i S˜t ∂y u(t, H ˜ t ∂x u(t, H ˜ t , S˜t ), β˜i H for i ≥ d + 1

˜ T − S˜T )+ ∈ L2 . Denoting Lt := For the specific case G(x, y) := (x − y)+ , one has X = (H ˜ t /S˜t , t ∈ [0, T ], gives X = S˜T (LT − 1)+ . A change of num´eraire dQ/d ˜ Q ¯ = e−˜µt S˜t /S˜0 H Ft   ˜ ˜t −σ ˜ S − 1 δ2t , yields π u (X) = eµ˜(T −t) S˜t E Q (LT − 1)+ . Now Lt = L0 e(α+ −˜µ)t exp β˜tr W ˜ tr W t

t

t

2

1/2 Pn P ˜2 1/2 , δ := |β| ˜ 2 + |˜ ˜ an nwith α± := γ˜ ± h σ |2 − 2 di=1 σ ˜i β˜i , and W i=d+1 βi /ai ˜ dimensional Q-Brownian motion. Now the formula (3.33) follows from the classical Margrabe formula for exchange options.

 √ Derivation of (3.35),(3.37). The stochastic exponential E (ε/ ν) · W ν is a uniformly in¯ ∈ Qngd ⊇ Qngd (see (3.4) for definition of tegrable martingale which defines a measure Q  √ √ e ¯ := ε/ ν, i.e. dQ/dP ¯ Qngd ⊂ M ) with Girsanov kernel λ = E (ε/ ν)·W ν . Indeed, applying √   √ [CFY05, Theorem 2.4 and Section 6] one gets that E (ε/ ν) · W ν and S = S0 E ν · WS ¯ is are uniformly integrable P - respectively Q-martingales. The variance process ν under Q

again a CIR process with parameters (¯ a, b, β, ρ) where a ¯ := a + βε 1 − ρ2 > a and the Feller ¯ condition β 2 ≤ 2¯ a still holds. For a put option X = (K − ST )+ ∈ L∞ , Y¯t := EtQ [X] are ¯ (instead of P ). Since the Heston given by the Heston formula (cf. [Hes93]), applied under Q price is non-decreasing in the mean reversion level of the variance process ([OA11, Proposition ¯ 5.3.1]) one expects that πtu (X) = Y¯t = EtQ [X]. Let us make this precise. For Q ∈ Qngd with √ Girsanov kernel λ satisfying |λ| ≤ ε/ ν, one has YTQ = Y¯T = X with YtQ = EtQ [X]. Using p

Section 3.4. Appendix

Page 142

Feynman-Kac, Y¯t = u(t, St , νt ) for a function u ∈ C 1,2,2 ([0, T ] × R+ × R+ ) with ∂u ∂ν ≥ 0 (see [OA11, Theorem 5.3.1, Corollary 5.3.1]). By Itˆo’s formula and change of measure follows q q √ √ ∂u ε  ∂u dY¯t =β 1 − ρ2 νt λt − √ (t, St , νt )dt + β 1 − ρ2 νt (t, St , νt )dWtQ,ν νt ∂ν ∂ν   √ ∂u √ ∂u (t, St , νt ) + βρ νt (t, St , νt ) dWtS , t ∈ [0, T ]. (3.80) + St νt ∂S ∂ν

Since X is bounded, then Y¯ is in S ∞ (Q) and a Q-supermartingale by (3.80) . Hence YtQ ≤ Y¯t for all Q ∈ Qngd , which by Part 1. of Theorem 3.7 implies the claim and thus we obtain the Heston type formula (3.35). ¯

¯ ∈ Qngd and π u (X) = E Q [X] with X ∈ L∞ , Corollary 3.10 implies that the good-deal Since Q 0 b of the ¯ ∈ S ∞ × H2 (note P = Q) bound is the Y -component of the minimal solution (Y¯ , Z) √ ¯ t z 2 = εz 2 / νt , for z = (z 1 , z 2 ), and terminal condition BSDE (3.36) with generator gt (z) = λ X. Now consider the strategy √ ∂u √ ∂u √ βρ φ¯t = Z¯t1 = St νt (t, St , νt ) + βρ νt (t, St , νt ) = St νt ∆t + Vt . ∂S ∂ν 2 Clearly φ¯ is in the set Φ = H2 (R) of permitted trading strategies since Z¯ ∈ H2 (R2 ). Recall that  √ P ngd consists of dQ/dP = E (λS , λν ) · W such that (λS , λν ) ≤ ε/ ν with (λS , λν ) being bounded. For Q ∈ P ngd , any wealth process φ · W S , φ ∈ Φ, is thus in S 1 (Q). As Qngd ⊆ P ngd R holds, clearly πtu (X) ≤ ρt (X − tT φs dWsS ) for any strategy φ ∈ Φ. To prove that φ¯ is a   R good-deal hedging strategy, we show the reverse inequality πtu (X) ≥ EtQ X − tT φ¯s dWsS for all Q ∈ P ngd . Let Q ∈ P ngd with Girsanov kernel (λS , λν ). Like in (3.80), we obtain for any stopping time τ that Y¯τ ∧T −

Z

τ ∧T

τ ∧t

φ¯s dWsS = Y¯τ ∧t +

Z

τ ∧T

τ ∧t

q √ ε  ∂u β 1 − ρ2 νs λνs − √ (s, Ss , νs )ds νs ∂ν

(3.81)

+ Lτ ∧T − Lτ ∧t , R p √ Q,ν for the local Q-martingale L := 0· β 1 − ρ2 νs ∂u ∂ν (s, Ss , νs )dWs . By the inequalities R √ τ ∧T ∂u ν S ¯ ¯ ¯ ∂ν ≥ 0 and λ ≤ ε/ ν follows that Yτ ∧T − τ ∧t φs dWs is less than Yτ ∧t + Lτ ∧T − Lτ ∧t . Localizing L along a sequence of stopping times τn ↑ ∞ and taking conditional Q-expectations R ∧T ¯ S ∈ S 1 (Q), the claim then yields EtQ [Y¯τn ∧T − ττnn∧t φ¯s dWsS ] ≤ Y¯τn ∧t . Using X ∈ L∞ and φ·W ∂u follows by dominated convergence. Hence (3.37) holds for Vt := ∂σ (t, St , νt ) = 2σt ∂u ∂ν (t, St , νt ) √ and volatility σt = νt .

Proof of Lemma 3.26. Part 1: We use (3.48) to show that the set θ∈Θ P ngd (P θ ) is m-stable and convex. Let κ ∈ [0, 1], τ ≤ T be a stopping time and Z i = E(λi · W 0 ), with λi selection S

Section 3.4. Appendix

Page 143

of C 0 + θi , θi ∈ Θ, i = 1, 2. From the proof of the second part of Lemma 3.1 the process Z := I[0,τ ] Z 1 + I]τ,T ] Z 2 Zτ1 /Zτ2 satisfies Z = E(λ · W 0 ) with λ = I[0,τ ] λ1 + I]τ,T ] λ2 . By convexity of the values of C 0 it follows that λ ∈ C 0 + θ for θ := I[0,τ ] θ1 + I]τ,T ] θ2 ∈ Θ S by convexity of the values of Θ. Hence Z is in P ngd (P θ ), and therefore θ∈Θ P ngd (P θ ) is m-stable. To show convexity, consider the density process Z˜ = κZ 1 + (1 − κ)Z 2 . Then 2 ˜ · W 0 ) with λ ˜ = 1 κZ 1 2 λ1 + (1−κ)Z Z˜ = E(λ λ2 . Again by convexity of the values κZ +(1−κ)Z κZ 1 +(1−κ)Z 2 2 2 ˜ ∈ C 0 + θ, ˜ for θ˜ := 1 κZ 1 2 θ1 + (1−κ)Z of C 0 , λ 1 2 θ ∈ Θ since Θ is convex-valued. κZ +(1−κ)Z

κZ +(1−κ)Z S Concerning Part 2: M-stability and convexity of θ∈Θ Qngd (P θ )   S S ngd (P θ ) T Me . 1, and θ∈Θ Qngd (P θ ) = θ∈Θ P

follow from that of Me , Part

4. Hedging under good-deal bounds and volatility uncertainty: a 2BSDE approach In this chapter, we study good-deal bounds defined from a bound on the instantaneous Sharpe ratios in the economy and a notion of robust hedging (as in Chapter 3) in the presence of volatility uncertainty. We describe worst-case good-deal bounds and robust hedging strategies in terms of solutions to 2BSDEs. In Section 4.1 we clarify the canonical setup incorporating volatility uncertainty and provide some preliminary results about 2BSDEs. Then in Section 4.2 we describe a model of the financial market under volatility uncertainty, together with a parametrization of the no-good-deal restriction in this model. Section 4.3 is devoted to the main results of the chapter, namely a 2BSDE characterization of good-deal bounds and associated hedging strategies and the fact that the latter are at least mean-self-financing uniformly over all priors (robustness). It includes in addition an example for European put options on non-tradeable assets in a Black-Scholes model with uncertain volatility, where worst-case valuations can be computed explicitly from a Black-Scholes’ type formula under a worst-case prior. Robust good-deal hedging strategies are also obtained in closed-form in this example, and it is shown that they are in general not super-replicating under volatility uncertainty.

4.1

Mathematical framework and preliminaries

We consider a canonical setting with filtered probability space (Ω, F, P 0 , F). Here Ω := {ω ∈ C([0, T ], Rn ) : ω(0) = 0} denotes the space of continuous paths starting at 0 and equipped with the norm kωk∞ := supt∈[0,T ] |ω(t)|. The canonical process B is defined by Bt (ω) := ω(t), for ω ∈ Ω and its law is P 0 , the Wiener measure. The underlying filtration F = (Ft )t∈[0,T ] is generated by B and F+ = (Ft+ )t∈[0,T ] denotes its right-limit, with Ft+ = Ft+ . For a probability measure Q, the conditional expectation given Ft will be denoted by EtQ [·].

4.1.1

The local martingale measures

A probability measure P is called a local martingale measure if B is a local martingale with respect to (F, P ). From [Kar95] (see also [F¨ol81]), it follows that there exists a F-progressively R R measurable process denoted by 0· Bstr dBs which coincides with the P -Itˆo integrals (P ) 0· Bs dBstr P -a.s. for all local martingale measures P . In particular, this yields path-wise definition of the R b with respect quadratic variation hBi of B as hBi := BB tr − 2 0· Bs dBstr and of its density a

144

Section 4.1. Mathematical framework and preliminaries

Page 145

to the Lebesgue measure dt as bt (ω) := lim sup a &0

hBit (ω) − hBit− (ω) , 

b is well-defined We denote by P W the set of all local martingale measures P for which a >0 n×n and takes values P -almost surely in the space Sn ⊂ R of positive definite symmetric n × n-matrices. As mentioned in [STZ11], the measures in P W can be typically mutually singular. In particular, there is no dominating measure in P W and this can be illustrated by the following example of [STZ11]: √  Example 4.1. For n = 1, P = P 0 , P 0 = P 0 ◦ ( 2B)−1 , A = hBit = t, for all t ∈ [0, T ]  and A0 = hBit = 2t, for all t ∈ [0, T ] , it holds P, P 0 ∈ P W , P (A) = P 0 (A0 ) = 1 and P (A0 ) = P 0 (A) = 0. Hence P ⊥P 0 . −1

bt 2 dBt , t ∈ [0, T ] is a Note that for any P ∈ P W , the process W P defined by WtP := (P ) 0t a P P Brownian motion under P (by L´evy characterization and since hW it = t, P -a.s.). Similarly to [STZ12], we will use the so-called strong formulation of volatility uncertainty according to which we consider only on the local martingale measures induced by the laws of solutions to 1/2 SDEs dXt = at (X)dBt , P -a.s.. More precisely, uncertainty will be considered only over the subclass P S ⊂ P W consisting of all probability measures R

α

0

α −1

P := P ◦ (X )

where

,

Xtα

Z

:= 0

t

αs1/2 dBs ,

P 0 -a.s. t ∈ [0, T ],

T with S>0 n −valued F-progressively measurable diffusion coefficient α satisfying 0 |αt |dt < ∞, P 0 -a.s.. The subscript S in P S stands for “strong” as in strong formulation, as opposed to W in P W which stands for “weak”. The consequence of restricting oneself to the subclass P S is the aggregation property it possesses in the sense that the following lemma (see [STZ11, Lemma 8.1, Lemma 8.2]) holds.

R

P

Lemma 4.2. For P ∈ P W , let F and FW P P FW .

P

denote respectively the P -augmentations of the P

P

filtrations F and Then P S = P ∈ P W : F = FW P , and B has the martingale representation property simultaneously with respect to all P ∈ P S . In addition, every P ∈ P S satisfies the Blumenthal Zero-One law. 

Remark 4.3. 1. For any P α ∈ P S one has P α ◦ B −1 = P 0 ◦ (X α )−1 , i.e. the distribution of B under P α coincides with the distribution of X α under P 0 . In particular with the filtration characterization of P S in Lemma 4.2, this implies that the density of the b(B) = α ◦ βα (B), P α ⊗ dt-a.s., for quadratic variation of B under P α is equal to a some F-progressively measurable map βα : [0, T ] × Ω → Rn (see [STZ13, Lemma 2.2])

Section 4.1. Mathematical framework and preliminaries

Page 146

2. Note that for any P ∈ P S , the Blumenthal Zero-One law in Lemma 4.2 implies EtP [X] = E P [X|Ft+ ] P -a.s. for any X ∈ L1 (P ), t ∈ [0, T ]. In particular, any Ft+ -measurable random variable has a Ft -measurable P -version. We will work with an even more restricted set of local martingale measures by considering for fixed a, a ∈ S>0 n , the subclass PH of P S defined by n

o

(4.1)

b ≤ a, P ⊗ dt-a.e. . PH = P ∈ P S : a ≤ a

κ (1 < κ ≤ 2) given Remark 4.4. The definition of PH is slightly different from the one for PH in [STZ12, Definition 2.6] as



κ b ≤ aP , P ⊗ dt-a.e., PH = P ∈ P S : ∃aP , aP ∈ S>0 n s.t. aP ≤ a

E

P

h Z

T

0

κ

2 i

bt )| dt |Ft (0, 0, a

κ



(4.2)

0 n ,

(4.3)

n×n is such that and we set Ft (ω, y, z, a) := +∞ for a ∈ Rn×n \ S>0 n , where DH ⊆ R n×n Ht (ω, y, z, γ) = +∞ for γ ∈ R \ DH , and DH is assumed to not-depend on (t, ω, y, z) and b bt ) and Fbt0 := Fbt (0, 0) for P ∈ PH . Let to contain the origin. We denote Ft (y, z) := Ft (y, z, a DFt (y,z) be the domain of F in a for fixed (t, ω, y, z). To obtain existence and uniqueness of solutions to 2BSDEs with generators F we need the following combination of Assumption 2.8 and Assumption 4.1 of [STZ12]:

Assumption 4.6. (i) PH = 6 ∅ and DFt (y,z) = DFt is independent of (ω, y, z), for all t ∈ [0, T ], (ii) F is F-progressively measurable in (t, ω) for any fixed (y, z, a), (iii) F is uniformly continuous in ω with respect to the supremum norm k · k∞ , (iv) Fb is PH -q.s. uniformly Lipschitz in (y, z), in the sense that ∃ C > 0 s.t. PH -q.s. for all y, y 0 ∈ R, z, z 0 ∈ Rn , 1/2

bt (z − z 0 )| , t ∈ [0, T ]. |Fbt (y, z) − Fbt (y 0 , z 0 )| ≤ C |y − y 0 | + |a

(v) Fb 0 satisfies

R

T b0 2 0 |Fs | ds

1/2

∈ L2H , i.e.

sup E P ∈PH



P

h

P

ess sup t∈[0,T ]

EtH,P

Z 0

T

|Fbs0 |2 ds

i

< ∞.

(4.4)

Remark 4.7. 1. Assumption 4.6, (iii) is less standard, and its importance lies in the proof of existence of solutions to 2BSDEs. In fact it provides the additional regularity of the generator needed to use the regular conditional probability distributions (shortly r.c.p.d.), which exist in the present canonical Wiener setting (see e.g. [SV79]). Using r.c.p.d. ensures a path-wise construction of solutions to 2BSDEs, i.e. without exception of negligible sets, hence avoiding any issue caused by singularity of measures in PH . 2. Assumption 4.6, (v) implies in particular that sup E P P ∈PH



Z 0

T

|Fbs0 |2 ds < ∞, 

(4.5)

2 in [STZ12]. Recall which in turn yields the integrability condition in the definition of PH that we have originally omitted (cf. Remark 4.4) this condition in the definition (4.1) of PH . For H such that Fb 0 is bounded PH -quasi-surely, (4.4) automatically follows.

Section 4.1. Mathematical framework and preliminaries

Page 149

Existence and uniqueness of solutions to 2BSDEs Following [STZ12], a second-order BSDE is a stochastic integral equation of the type Yt = X −

Z

T

Z

Fbs (Ys , Zs )ds −

t

t

T

Zstr dBs + KT − Kt ,

t ∈ [0, T ], PH -q.s.,

(4.6)

or equivalently −dYt = −Fbt (Yt , Zt )ds − Zttr dBt + dKt ,

t ∈ [0, T ],

YT = X, PH -q.s..

The solution to the 2BSDE (4.6) is defined as follows. Definition 4.8. For X ∈ L2H , a couple (Y, Z) ∈ D2H × H2H is called solution to the 2BSDE (4.6) if YT = X PH -q.s., the process K P defined for each P ∈ PH by KtP := Y0 − Yt +

t

Z

Z

Fbs (Ys , Zs )ds + 0

0

t

Zstr dBs ,

t ∈ [0, T ], P -a.s..

(4.7)

is P -a.s. non-decreasing, and the family K P , P ∈ PH satisfies the minimum condition 

KtP =

P

ess inf

0

0

P 0 ∈PH (t+ ,P )

EtP [KTP ],



P -a.s., for all P ∈ PH , t ∈ [0, T ].

(4.8)

If moreover the family {K P , P ∈ PH } can be aggregated into a universal process K, i.e. K = K P , P -a.s. for all P ∈ PH (see [STZ11] for more on aggregation), then one calls (Y, Z, K) solution to the 2BSDE. The pair (F, X) will be called the parameters (generator and terminal condition) of the 2BSDE (4.6). Y will be referred to as value process and Z as the control process. The following proposition is a combination of [STZ12, Theorem 4.3, Theorem 4.6], and provides conditions for existence and uniqueness of solutions to 2BSDEs with Lipschitz generators. In addition it gives a representation of the value process of the 2BSDE in terms of the value processes (under P ∈ PH ) of the associated standard BSDEs. We employ the classical notation for standard BSDEs (as in e.g. [EPQ97]) according to which the generator of the BSDE (4.10) is −Fb (i.e. with a minus sign). The notation for the generator of the associated 2BSDEs however remains unchanged. Proposition 4.9. Let Assumption 4.6 hold. Then 1. Assume that X ∈ L2H and that (Y, Z) ∈ D2H × H2H is a solution to the 2BSDE (4.6). Then for any P ∈ PH , Y has the representation Ys =

P

ess sup

P 0 ∈PH (s+ ,P )

0

YsP (t, Yt ), P -a.s., s ≤ t ≤ T,

(4.9)

Section 4.1. Mathematical framework and preliminaries

Page 150

where for each P ∈ PH , the couple (Y P (τ, ξ), Z P (τ, ξ)) is the unique solution to the standard BSDE with parameters (−Fb , ξ): YtP = ξ −

Z t

τ

Fbs (YsP , ZsP )ds −

Z t

τ

(ZsP )tr dBs ,

t ≤ τ, P -a.s.,

(4.10)

for a F+ -stopping time τ and Fτ+ -measurable random variable ξ ∈ L2 (P ). In particular, the 2BSDE (4.6) has at most one solution in D2H × H2H . 2. For X ∈ L2H , the BSDE (4.6) admits a unique solution (Y, Z) ∈ D2H × H2H . Remark 4.10. 1. From the dynamics of the value process Y , the control process Z of the 2BSDE (4.6) is uniquely given by dhY, Bit = Zt dhBit , PH -q.s.. As a consequence, one can obtain Z from Y as Zt = lim sup &0

hY, Bit − hY, Bit− , hBit − hBit−

t ∈ [0, T ].

2. Note that the representation (4.9) in Proposition 4.9 naturally follows from the minimum condition (4.8); this is a key step in deriving the probabilistic representation of solutions to fully nonlinear PDEs via 2BSDEs. With this at hand, uniqueness of the solution to the 2BSDE is a direct consequence of uniqueness for standard BSDEs. 3. As can be seen in part 2 of Proposition 4.9, sufficient conditions for existence and uniqueness of solutions to Lipschitz 2BSDEs crucially rely on the terminal X being in L2H . Clearly a trivial example of random variables X that lie in L2H are those in UCb (Ω), i.e. that are uniform continuous and bounded. These include e.g. constants and also random variables that can be written as X := g(Bt1 , . . . , Btk ), with t1 , . . . , tk ∈ [0, T ] and some bounded uniformly continuous function g : Rn×k → R, k ∈ N. This is true because since the function ω → ω(t) is Lipschitz continuous in the norm k·k∞ for any t ∈ [0, T ], then X would be uniformly continuous as it is the composition of uniformly continuous functions. In particular, it is sufficient for this purpose that the function g be Lipschitz and bounded.

Comparison theorems for 2BSDEs As theoretical results for 2BSDEs in this chapter, we now state and prove some comparison theorems for 2BSDEs with different generators. Unlike the well-known comparison theorem for standard BSDEs, and because of the presence of the non-decreasing processes in the 2BSDE-dynamics, comparison of the generators of the 2BSDEs at one of the solutions does not suffice to imply comparison of the value processes. We distinguish two approaches: The first one leads to Proposition 4.12 and assumes that the generators of the two 2BSDEs at the

Section 4.1. Mathematical framework and preliminaries

Page 151

solutions of one of the associated standard BSDEs are quasi-surely comparable. The second approach (cf. Theorem 4.13) rather assumes that the generators at the solution of one of the 2BSDEs are comparable and that an additional monotonicity condition on the difference of the non-decreasing processes holds, and obtain partly as a result that this difference also satisfies the minimum condition (4.8). For consistency with the current setup, our results are stated with respect to the family PH of mutually singular measures, but the proofs would be κ of [STZ12] as defined in (4.2). For two analogous for the more general families of measures PH 2BSDEs with the same generator but different terminal conditions, a comparison theorem was stated in [STZ12, Corollary 4.4] as a by-product of the representation result (4.9). However in applications, one can sometimes be concerned with 2BSDEs with different generators. In [PZ13, Proposition 3.1], a comparison theorem is proved (as generalization of [Tev08, Theorem 2]) assuming that the difference of the non-decreasing components is also non-decreasing. Their focus is on quadratic 2BSDEs, but they also state the result for 2BSDEs with same generators. Here we prove general comparison theorems for 2BSDEs with possibly different generators. Our proofs rely on a classical linearization procedure (also used in [STZ12]) coupled with a change of measure argument. Instead of imposing specific conditions on the generators which imply existence of solutions, we only insist that we have solutions and impose conditions on the generators and other processes of interest quasi-surely. To this end, the following intermediate result will be needed. Lemma 4.11. Let X ∈ L2H , λ and η be bounded F-progressively measurable R- and Rn -valued processes respectively and ϕ ∈ H2H . Let (Y, Z) ∈ D2H × H2H be a solution to the BSDE Z

Yt = X + t

T

 b1/2 ϕs + λs Ys + ηstr a s Zs ds −

Z t

T

Zstr dBs + KTP − KtP , t ∈ [0, T ], P -a.s., (4.11)

with K P nondecreasing and K0P = 0, for all P ∈ PH . If X ≥ 0 and ϕt ≥ 0, t ∈ [0, T ], P -a.s. for all P ∈ PH and, then Yt ≥ 0, t ∈ [0, T ], P -a.s. for all P ∈ PH . If in addition Y0 = 0, then X = 0, ϕt = 0 and Yt = 0, t ∈ [0, T ], P -a.s. for all P ∈ PH . R

−1/2

bs Proof. Let P ∈ PH and M be defined by Mt := exp 0t ηstr a dBs + P -a.s., t ∈ [0, T ]. Applying Itˆo’s product rule between t and T gives

Mt Y t = MT X −

Z t

T

−1

bs 2 ηs )tr dBs + Ms (Zs + Ys a −1

Z t

T

Ms dKsP +

Z t

T

Rt



1 2 0 (λs + 2 |ηs | )ds ,

ϕs Ms ds, P -a.s.. (4.12)

bs 2 )tr dBs is a P -martingale. Indeed, under P one can The process N := 0· Ms (Zs + Ys ηs a Rt 1/2 bs Zs + Ys ηs )tr dWsP , t ∈ [0, T ], where W P is a P -Brownian motion. write Nt = 0 Ms (a Hence (BDG) inequality it suffices to show that the P -expectation h Rby Burkholder-Davis-Gundy 1/2 i 1/2 T P 2 2 b E is finite. Because λ and η are bounded, Z ∈ H2H and 0 Ms |as Zs + Ys ηs | ds R

Section 4.1. Mathematical framework and preliminaries

Page 152

Y ∈ D2H , it follows by BDG inequality that E

P

h Z

T

0

2 b1/2 Ms2 |a s Zs + Ys ηs | ds

h

≤ E P sup Ms

Z

T

2 b1/2 |a s Zs + Ys ηs | ds

0

s≤T

≤ E P sup Ms2 

1/2

EP

1/2



hZ



T

0

s≤T

≤ E P sup Ms2

1/2 i

EP



2 b1/2 |a s Zs + Ys ηs | ds T

Z 0

s≤T

1/2 i

2 b1/2 |a s Zs | ds

1/2

i1/2

+ kηk∞ T 1/2 E P sup |Ys |2 

1/2 

< ∞,

s≤T

where the second and third inequalities are obtained using H¨older’s and Minkowski’s inequalities respectively. Therefore N is a true P -martingale and taking the conditional expectation in (4.12) yields Yt =

 Mt−1 EtP MT X

T

Z

+ t

Ms dKsP

T

Z



ϕs Ms ds ,

+ t

P -a.s., for all P ∈ PH .

(4.13)

Now if X ≥ 0 and ϕ ≥ 0, then it follows from (4.13) that Y ≥ 0 (since M > 0 and K P is non-decreasing). If moreover Y0 = 0 then E

P

Z

MT X + 0

T

Ms dKsP

Z

T



ϕs Ms ds = 0,

+ 0

for all P ∈ PH .

(4.14)

Finally since the random variable inside the expectation in (4.14) is non-negative and M > 0, then X = 0, ϕ = 0 and K P = 0. Therefore Y = 0. Note n that the proofoof Lemma 4.11 does not require the minimum condition (4.8) to be satisfied for K P , P ∈ PH . Lemma 4.11 will be used to prove the second comparison Theorem 4.13, the first being the following Proposition 4.12. Let X i be in L2H and F i be the generator associated by (4.3) to a nonlinear function H i (for i = 1, 2) and satisfying (4.5) and Assumption 4.6, (i),(ii),(iv). Let (Y i , Z i ) ∈ D2H × H2H be a solution to the 2BSDE with parameters (F i , X i ), having the representation (4.9). Suppose X1 ≥ X2

and

Fbt1 Yt2,P , Zt2,P ≤ Fbt2 Yt2,P , Zt2,P , P -a.s., for all t ∈ [0, T ], P ∈ PH , 



where (Y i,P , Z i,P ) denotes the solution of the standard BSDE with parameters (−Fb i , X i ) under P , for P ∈ PH (for i = 1, 2). Then Yt1 ≥ Yt2 , t ∈ [0, T ], P -a.s. for all P ∈ PH . Proof. Applying the comparison principle [EPQ97, Theorem 2.2] for standard BSDEs, one obtains Yt1,P ≥ Yt2,P , P -a.s., for all t ∈ [0, T ], for all P ∈ PH . Now for any fixed P ∈ PH ,

Section 4.1. Mathematical framework and preliminaries

Page 153

taking the essential supremum over all P 0 ∈ PH (t+ , P ) yields Yt1 =

P

ess sup

P 0 ∈P

H

(t+ ,P )

Yt1,P

0 ,X 1



P

ess sup

P 0 ∈P

H

(t+ ,P )

Yt2,P

0 ,X 2

= Yt2 ,

P -a.s., t ∈ [0, T ],

which proves the required result by using (4.9). If an hypothesis on Fb 1 and Fb 2 as in Proposition 4.12 is satisfied PH -quasi-surely at (Y 2 , Z 2 ),  instead for all Y 2,P , Z 2,P , P ∈ PH , then by imposing an additional monotonicity condition on the differences of the non-decreasing components of the associated 2BSDEs, we obtain the following similar result. Theorem 4.13. Let X i be in L2H and F i be the generator associated to a nonlinear function H i (i = 1, 2) and satisfying (4.5) and Assumption 4.6, (i),(ii),(iv). Let (Y i , Z i ) ∈ D2H × H2H be a solution to the 2BSDE with parameters (F i , X i ). Suppose X 1 ≥ X 2,

Fbt1 (Yt2 , Zt2 ) ≤ Fbt2 (Yt2 , Zt2 ), for all t ∈ [0, T ], P -a.s. for all P ∈ PH ,

and K 1,P − K 2,P is non-decreasing for all P ∈ PH , where

n

K i,P , P ∈ PH

o

are the non-

(F i , X i ),

decreasing processes associated to the 2BSDEs i = 1, 2. Then the minimum condition  1,P 2,P 1 holds for the family K − K , P ∈ PH , and Yt ≥ Yt2 , t ∈ [0, T ], P -a.s. for all P ∈ PH . Proof. Let δY = Y 1 − Y 2 , δZ = Z 1 − Z 2 and δK = K 1 − K 2 . Then using Assumption 4.6, iv) on F 1 and the classical linearization technique, one can construct λ, η two bounded, F-progressively measurable processes valued in R and Rn respectively such that for all t ∈ [0, T ] it holds P -a.s for any P ∈ PH that δYt = (X 1 − X 2 ) +

Z

T

b1/2 δ2 Fbs + λs δYs + ηstr a s δZs ds − 

t

Z

T

t

δZstr dBs + δKTP − δKtP ,

where δ2 Fbt = Fbt2 (Yt2 , Zt2 ) − Fbt1 (Yt2 , Zt2 ) ≥ 0, t ∈ [0, T ], P -a.s. By assumption, the process n o P 1,P 2,P P δK := K −K is non-decreasing and starts at 0. Moreover, δK , P ∈ PH also satisfies the minimum condition (4.8). Indeed let P ∈ PH and t ∈ [0, T ], then for all P 0 ∈ PH (t+ , P ) holds 0

0

0

0

0

0

0

δKtP = δKtP ≤ EtP [δKTP ] = EtP [KT1,P ] − EtP [KT2,P ], P -a.s.. Taking the essential infimum over all P 0 ∈ PH (t+ , P ) on both sides yields δKtP ≤

ess inf

P

P 0 ∈PH (t+ ,P )

0

0

EtP [δKTP ] = ≤

ess inf

P

P 0 ∈PH (t+ ,P )

ess inf

P

P 0 ∈PH (t+ ,P )



0

0

0

0



EtP [KT1,P ] − EtP [KT2,P ] , 0

0

EtP [KT1,P ] −

ess inf

P

P 0 ∈PH (t+ ,P )

P -a.s.. 0

0

EtP [KT2,P ],

P -a.s.,

Section 4.1. Mathematical framework and preliminaries

Page 154 n

which by the minimum condition on K 1,P , P ∈ PH and K 2,P , P ∈ PH 

δKtP ≤

ess inf

P 0 ∈P

H

P



0

(t+ ,P )

0

EtP [δKTP ] ≤ Kt1,P − Kt2,P = δKtP

o

yields

P -a.s..

This implies that δK P , P ∈ PH satisfies the minimum condition. By assumptions on F 1 , F 2 it clearly holds δ2 Fb ∈ H2H . Now since δ2 Fb ≥ 0 and X1 − X2 ≥ 0, then Lemma 4.11 implies δY ≥ 0. 



The following are direct consequences of Proposition 4.12 and Theorem 4.13, that could be used to describe in terms of 2BSDEs the solution to optimization problems that are stated with respect to mutually singular measures in PH . Corollary 4.14. Let X, X ϑ ∈ L2H and F, F ϑ associated to nonlinear functions H, H ϑ satisfying (4.5) and Assumption 4.6, (i),(ii),(iv), for ϑ in some index set Θ. Let (Y, Z), (Y ϑ , Z ϑ ) in D2H × H2H be solutions to the 2BSDEs with parameters (F, X), (F ϑ , X ϑ ) and having the representation (4.9). Suppose there exists ϑ¯ ∈ Θ such that P

¯

X = ess inf X ϑ = X ϑ , ϑ∈Θ

P -a.s., P ∈ PH

P ¯ Fbt (YtP , ZtP ) = ess sup Fbtϑ (YtP , ZtP ) = Fbtϑ (YtP , ZtP ),

and P -a.s., t ∈ [0, T ], P ∈ PH ,

ϑ∈Θ

where (Y P , Z P ), (Y ϑ,P , Z ϑ,P ) denote the solutions under P to the standard BSDEs with P

¯

parameters (−Fb , X), (−Fb ϑ , X ϑ ) respectively. Then Yt = ess inf Ytϑ = Ytϑ P -a.s., for all ϑ∈Θ

t ∈ [0, T ], P ∈ PH . ¯

Proof. By the hypotheses on the generators, follows Y P = Y ϑ,P , P -a.s., P ∈ PH . This ¯ ¯ implies by the representation (4.9) that Y = Y ϑ . Now Proposition 4.12 yields Ytϑ = Yt ≤ Ytϑ , P -a.s. for all t ∈ [0, T ], P ∈ PH , for all ϑ ∈ Θ. After taking the essential infimum over all ϑ ∈ Θ, this yields P

P

¯

ess inf Ytϑ ≤ Ytϑ = Yt ≤ ess inf Ytϑ P -a.s., for all t ∈ [0, T ], P ∈ PH , ϑ∈Θ

ϑ∈Θ

which is the required result. Corollary 4.15. Let X, X ϑ ∈ L2H and F, F ϑ associated to nonlinear functions H, H ϑ satisfying Assumption 4.6, for ϑ in some index set Θ. Let (Y, Z), (Y ϑ , Z ϑ ) ∈ D2H × H2H be solutions to the 2BSDEs with parameters (F, X), (F ϑ , X ϑ ). Suppose there exists ϑ¯ ∈ Θ such that P

¯

X = ess inf X ϑ = X ϑ P -a.s., for all P ∈ PH , ϑ∈Θ

P ¯ Fbt (Yt , Zt ) = ess sup Fbtϑ (Yt , Zt ) = Fbtϑ (Yt , Zt ), P -a.s., for all t ∈ [0, T ], P ∈ PH , ϑ∈Θ

Section 4.2. Market model and good-deal constraint under volatility uncertainty and K ϑ,P − K P is non-decreasing for all ϑ ∈ Θ, P ∈ PH , where n

K ϑ,P , P ∈ PH

o

Page 155

n

K P , P ∈ PH

o

and

are the non-decreasing components of the solutions (Y, Z) and (Y ϑ , Z ϑ )

for the 2BSDEs with parameters (F, X) and (F ϑ , X ϑ ). Then for all t ∈ [0, T ], P ∈ PH , one has P

P

¯

Yt = ess inf Ytϑ = Ytϑ ϑ∈Θ

¯

KtP = ess inf Ktϑ,P = Ktϑ,P , P -a.s..

and

ϑ∈Θ

Proof. By the hypotheses on the generators it follows for all P ∈ PH and t ∈ [0, T ] that KtP = Y0 − Yt + = Y0 − Yt +

t

Z 0

0 t

Z 0

¯

Since X = X ϑ and

n

t

Z

Fbs (Ys , Zs )ds +

K P , P ∈ PH

o

¯

Fbsϑ (Ys , Zs )ds +

Zstr dBs , P -a.s.

t

Z 0

(4.15)

Zstr dBs , P -a.s..

satisfies the minimum condition, then by uniqueness ¯

¯

¯

of solutions to 2BSDEs, (4.15) implies (Y, Z) = (Y ϑ , Z ϑ ). This yields KtP = Ktϑ,P , P -a.s., t ∈ [0, T ], for all P ∈ P. Moreover by Theorem 4.13 one obtains from the hypotheses on the generators and the associated non-decreasing processes that Yt ≤ Ytϑ holds P -a.s., for all t ∈ [0, T ], P ∈ PH , for all ϑ ∈ Θ. Now taking the essential infimum over all ϑ ∈ Θ yields P

P

¯

ess inf Ytϑ ≤ Ytϑ = Yt ≤ ess inf Ytϑ ϑ∈Θ

ϑ∈Θ

P -a.s., for all t ∈ [0, T ], P ∈ PH .

Furthermore, observe that since for all ϑ ∈ Θ the process K ϑ,P − K P is non-decreasing P -a.s., then the process

P

ess inf K ϑ,P − K P is also P -a.s. non-decreasing and starts at 0 for all ϑ∈Θ

¯

P ∈ PH . In addition, since K P = K ϑ,P , for all P ∈ PH , then the inequalities P

P

¯

0 ≤ ess inf Ktϑ,P − KtP ≤ ess inf Ktϑ,P − K ϑ,P ≤ 0. ϑ∈Θ

ϑ∈Θ

hold.

4.2

Market model and good-deal constraint under volatility uncertainty

We apply the preceding 2BSDE theory to good-deal valuation and hedging of contingent claims in incomplete financial markets under volatility uncertainty. Recall (cf. [DM06, DK13a, NS12, EJ13, EJ14, Vor14]) that in the framework of volatility uncertainty, the reference probability measures interpreted as generalized scenarios in the market (cf. [ADE+ 07]) are no longer dominated and may actually be mutually singular. In comparison to standard BSDEs which are

Section 4.2. Market model and good-deal constraint under volatility uncertainty

Page 156

used in Chapters 3 and 2 in the presence of drift uncertainty or absence of uncertainty at all, 2BSDEs seem to be an appropriate tool for describing worst-case valuations in the presence of volatility uncertainty (see also [MPZ15]). We will characterize worst-case good-deal bounds and associated robust hedging strategies via solutions to 2BSDEs. As in [CR00, BS06], we consider good-deal constraints imposed as bounds on the Sharpe ratios (equivalently bounds on the optimal growth rates as in [Bec09]) in the financial market extended by additional wealth processes. First let us specify the model for the market with uncertainty about the volatility.

4.2.1

Financial market with volatility uncertainty

The financial market consists of d tradeable stocks (d ≤ n) with discounted price processes (S i )di=1 = S modelled by dSt = diag(St )σt dBt , t ∈ [0, T ], PH -q.s.,

S0 ∈ (0, ∞)d ,

(4.16)

where σ is a Rd×n -valued F-predictable process, each σt being uniformly continuous in ω with respect to k · k∞ . We assume that σσ tr is uniformly bounded and uniformly elliptic, i.e. there exists K, L > 0 such that

K Id ≤ σσ tr ≤ L Id ,

PH ⊗ dt-q.s.,

(4.17)

b1/2 is PH ⊗ dt-q.s. of maximal rank where Id denotes the d × d identity matrix. In particular σ a tr bt σt is uniformly elliptic and bounded (using (4.1) and (4.17)). d ≤ n, since σt a

Remark 4.16.

1. From (4.17) and (4.1) holds supP ∈PH E P

 RT σt a1/2 2 dt < ∞, and 0 n R o (P )

σs dBs , P ∈ PH

hence by [DM06, Lemma 2.4 and Theorem 2.8] the family

of



stochastic integrals can be aggregated into a single process 0 σs dBs that is defined PH -quasi-surely. In fact under additional assumptions (e.g. c`adl`ag integrands as in [Kar95], or continuum hypothesis as in [Nut12b]) they can even be defined path-wise without exception of a null-set. 2. The market model captures uncertainty about the volatility in the sense that under each 1/2 bt dWtP , where W P is a P -Brownian motion. In measure P ∈ PH , one has dBt = a fact, substituting this in the dynamics of S in (4.16) one sees that under the reference b1/2 plays the role of the instantaneous volatility matrix measure P ∈ PH , the process σ a for the stock prices S. In this sense, Knightian uncertainty (ambiguity) about future volatility scenarios is captured by the local martingale laws P ∈ PH for S. 3. The bounds a, a ¯ and the uniform bounds on σσ tr can be viewed as setting a confidence region for future volatility values, calibrated e.g. from extreme implied (or historical) volatilities in the market.

Section 4.2. Market model and good-deal constraint under volatility uncertainty

Page 157

4. The financial market described is incomplete under any scenario P ∈ PH for the volatility b1/2 if d < n, since σ a b1/2 is of full rank PH ⊗ dt-q.s.. σa Let Me (P ) be the set of equivalent local martingale measures of S under each model P , for  P ∈ PH . Denoting (P )E(M ) := exp M − M0 − 12 hM iP the stochastic exponential of the local P -martingale M under P , we have the following Lemma 4.17. For P ∈ PH , the set Me (P ) consists of the equivalent measures Q ∼ P such 1/2 bt ), t ∈ [0, T ]. that dQ = (P )E(η · W P )dP with η F-progressively measurable and ηt ∈ Ker (σt a Proof. Let P ∈ PH . By the martingale representation theorem under P (see Lemma 4.2), any Q ∈ Me (P ) satisfies dQ = (P )E(η · W P )dP for a F-progressive measurable process η such R that 0T |ηs |2 ds < ∞, P -a.s. holds. By Girsanov theorem, one can rewrite the dynamics of S  R 1/2 1/2 bt dWtQ , t ∈ [0, T ], where W Q = W P − 0· ηs ds bt ηt dt + σt a under Q as dSt = diag(St ) σt a is a Q-Brownian motion by Levy’s characterization. Now Q is a local martingale measure for S 1/2 1/2 bt ηt = 0 for all t ∈ [0, T ], i.e. if and only if ηt ∈ Ker (σt a bt ), t ∈ [0, T ]. if and only if σt a Remark 4.18. 1. Note from (4.16) that the measures P ∈ PH are also local martingale measures for S. This implies that P ∈ Me (P ) 6= ∅, for any P ∈ PH . As a consequence, the market satisfies the no-free lunch with vanishing risk condition (see [DS94]) under each P ∈ PH . This is equivalent to a robust notion for no-arbitrage under uncertainty (see [BBKN14]). 2. Modeling the stock price process directly under local-martingale measures (i.e. setting its drift to zero) is a technical assumption rather than financially justified. In our case, this will ensure convexity (in a ∈ S>0 n ) of the generators F of the upcoming pricing and hedging 2BSDEs. This convexity is essential in 2BSDE theory since F is defined by (4.3) as the convex conjugate of a function H. Confer part 1 of Remark 4.21 for further notes about the possible limitations for the applicability of 2BSDE theory if one includes a non-zero drift in (4.16) . We parametrize trading strategies ϕ = (ϕi )di=1 in terms of the amount ϕi of wealth invested in the stock with price process S i , with ϕ being a F+ -progressively measurable process with suitable integrability properties. In this respect, the wealth process V ϕ associated to a trading strategy ϕ with initial capital V0 (so that (V0 , ϕ) quasi-surely satisfies the self-financing requirement) has the dynamics Vtϕ = V0 +

Z 0

t

ϕtr s σs dBs ,

t ∈ [0, T ], PH -q.s..

Section 4.2. Market model and good-deal constraint under volatility uncertainty

Page 158

Re-parameterizing trading strategies in terms of integrands φ := σ tr ϕ ∈ Im σ tr with respect to B, the dynamics of the wealth process V φ := V ϕ rewrites Vtφ = V0 +

Z 0

t

φtr s dBs = V0 +

t

Z

(P )

0

1

bs2 dWsP , P -a.s., t ∈ [0, T ], P ∈ PH . φtr sa

(4.18)

We denote Φ(P ), P ∈ PH the set of trading strategies that are permitted under P (referred to as P -permitted), defined as n

Φ(P ) := φ : φ is F -prog. meas., E +

P

Z

T

0

o

tr 2 tr b1/2 |a , s φs | ds < ∞, and φ ∈ Im σ



with “prog. meas.” abbreviating progressively measurable. We use the following definition of the set of permitted trading strategies. Definition 4.19. The set Φ of permitted trading strategies under volatility uncertainty consists of all F+ -progressively measurable processes φ ∈ Im σ tr satisfying sup E P



Z

P ∈PH

0

T

tr 2 b1/2 |a s φs | ds < ∞

and such that the family of stochastic integrals single process



n

R (P ) · φtr dB , s 0 s

P ∈ PH

o

aggregates into a



tr 0 φs dBs .

· tr By its quasi-sure definition, the integral 0· φtr s dBs , for a strategy φ ∈ Φ, satisfies 0 φs dBs = R · (P ) φtr dB , P -a.s. for all P ∈ P . The trading strategies in Φ are termed as P -permitted s H H 0 s T (or simply permitted). Clearly V φ is a P -martingale for any φ ∈ Φ ⊆ P ∈PH Φ(P ) and b1/2 of the P ∈ PH , hence excluding existence of arbitrage strategies in Φ for any scenario σ a volatility.

R

4.2.2

R

No-good-deal constraint

In the absence of uncertainty, we consider a no-good-deal constraint defined as a bound on the instantaneous Sharpe ratios, for any market extension by additional derivative price processes obtained from the no-good-deal pricing measures (cf. [CR00, BS06] and references therein). This no-good-deal constraint is equivalent to a bound on the optimal expected growth rates of returns, again in any market extension (see [Bec09]). Classically, such can be ensured (using the Hansen-Jagannathan inequality) by imposing a bound on the norm of Girsanov kernels for risk-neutral pricing measures. In the presence of drift (rather than volatility) uncertainty, results about good-deal valuation and robust hedging are provided in Chapter 3. Our aim here is to derive analogs of these results in the presence of volatility uncertainty. The no-good-deal constraint under volatility uncertainty consists of imposing the same bound h on the Girsanov

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 159

kernels of pricing measures in every model P ∈ PH separately. By doing this, we obtain for each P a set of no-good-deal measures Qngd (P ) ⊆ Me (P ). Following a worst-case approach to good-deal valuation under uncertainty (as in (3.49) in Chapter 3, but taking into account here the possible singularity of the priors P ∈ PH ), this will yield a larger good-deal bound obtained as the supremum of prices taken over all no-good-deal measures for all reference measures P ∈ PH . To be more precise let h be a fixed positive bounded F-progressively measurable process that is uniformly continuous in ω with respect to k · k∞ . We consider the set Qngd (P ) of no-good-deal measures in the model P ∈ PH as the subset of Me (P ) consisting of equivalent local martingale measures Q, whose Girsanov kernels η with respect to the P -Brownian motion W P are bounded by h, i.e. |ηt (ω)| ≤ ht (ω) for all (t, ω) ∈ [0, T ] × Ω. In other words, using Lemma 4.17, we define n







Qngd (P ) := Q ∼ P dQ/dP = (P )E η · W P , with F-prog. meas. η o

b1/2 ) and |η| ≤ h . satisfying η ∈ Ker (σ a

Clearly, for all P ∈ PH holds P ∈ Qngd (P ) 6= ∅. Note that uniform continuity of h and σ will ensure that the forthcoming 2BSDE generators satisfy Assumption 4.6, iii), needed for wellposedness of the associated 2BSDEs (see Theorem 4.9). As in part b) of Lemma 3.1 in Chapter 3, one can show that for P ∈ PH the set Qngd (P ) is convex and multiplicatively stable (in short m-stable). M-stability of a set of priors is usually key for obtaining time-consistency of the corresponding process dynamically defined as essential supremum over conditional expectations over the priors; see [Del06] for the definition and a general study of m-stability when the priors are dominated. M-stability is also referred to as rectangularity in the economic literature [CE02].

4.3

Good-deal bounds and hedging under volatility uncertainty

Using 2BSDEs, we describe good-deal bounds in the market model of Section 4.2.1 and study an associated notion of robust hedging in the framework of volatility uncertainty. We first define the good-deal valuation bounds whose financial motivation comes from the no-good-deal restriction mentioned previously. Then we characterize the corresponding good-deal bounds in terms of solutions to Lipschitz 2BSDEs. After that, we derive hedging strategies as minimizers of some dynamic coherent a-priori risk measure ρ under volatility uncertainty (e.g. as in [NS12]), so that the good-deal bound arises as the market consistent risk measure associated to ρ, in the spirit of [BE09]. Our definition of the good-deal bounds and hedging strategies will take into account the dependence of the no-good-deal restriction on the prior, and the aversion of investors to volatility uncertainty.

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

4.3.1

Page 160

Good-deal bounds under volatility uncertainty

As in Chapter 3, Section 3.3, the main idea behind good-deal valuation under uncertainty is to view aversion to model uncertainty as a penalization of the no-good-deal restriction yielding a larger good-deal bound than in the absence of uncertainty. We use a worst-case approach to uncertainty aversion in the spirit of [GS89, HS01, CE02]. This approach has been used for example in [ALP95, Lyo95, NS12, Vor14] to study robust arbitrage bounds and super-hedging strategies in a financial market with volatility uncertainty or in [SW05, Sch07, Que04, DK13a, MPZ15] for robust utility maximization under model uncertainty. Intuitively, an uncertainty-averse investor faced with insufficient knowledge about the actual financial market volatility, would opt for a worst-case approach to valuation in order to compensate for eventual losses due to the wrong choice of the volatility. Acting this way, she would sell (resp. buy) financial risks at the largest (resp. smallest) good-deal bounds over all possible scenarios in her confidence set of volatility values, corresponding to the set PH of reference priors. Acknowledging that mutual singularity of the reference measures in PH brings additional technical difficulties in making rigorous sense of essential suprema, we define the (robust) worst-case good-deal bound π·u (X) in our dynamic framework for a financial risk X ∈ L2H as the unique process π·u (X) ∈ D2H (if it exists) that satisfies πtu (X) =

P

ess sup

P0

ess sup EtQ [X], t ∈ [0, T ], P -a.s., for all P ∈ PH .

P 0 ∈PH (t+ ,P ) Q∈Qngd (P 0 )

(4.19)

The definition of the lower good-deal bound π·l (X) = −π·u (−X) is analogous, replacing the essential suprema in (4.19) by essential infima; for this reason we focus only on studying the upper bound. For X ∈ L2H , the good-deal bound π·u (X) will be shown to be a single universal process corresponding to the Y -component of the solution of a 2BSDE. Before proceeding, let us introduce some notations that will be used throughout the sequel. a,⊥ a For a ∈ S>0 n , we denote by Πt (·) and Πt (·) respectively the orthogonal projections onto the subspaces Im (σt a1/2 )tr and Ker (σt a1/2 ) of Rn , t ∈ [0, T ]. More precisely for each a ∈ S>0 n n and t ∈ [0, T ], we define the projections of z ∈ R as Πat (z) = (σt a1/2 )tr (σt aσttr )−1 (σt a1/2 )z

a and Πa,⊥ t (z) = z − Πt (z).

(4.20)

b at ,⊥ at b t (·) := Πb b⊥ In particular we define (in a path-wise sense) Π (·). For each t (·) and Πt (·) := Πt 0 + P ∈ PH , t ∈ [0, T ] and P ∈ PH (t , P ), the standard good-deal bound in the model P 0 is 0 0 given as usual by πtu,P (X) := ess supPQ∈Qngd (P 0 ) EtQ [X], P -a.s., so that by (4.19) one has

πtu (X) =

P

ess sup

P 0 ∈PH (t+ ,P )

0

πtu,P (X), P -a.s., t ∈ [0, T ], P ∈ PH , for X ∈ L2H . 0

(4.21)

Note from Theorem 3.15 in Chapter 3 that the good-deal bound π·u,P (X) for P 0 ∈ PH (t+ , P ) and P ∈ PH is the value process of the standard BSDE under P with generator −Fbt (·) =

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 161

bt ), t ∈ [0, T ], and terminal condition X, with F given for z ∈ Rn , a ∈ Rn×n by −Ft (·, a  −h Πa,⊥ a 12 z  t t F (t, z, a) = +∞

if a ∈ S>0 n and a ≤ a ≤ a,

(4.22)

otherwise.

For X ∈ L2H , we consider the 2BSDE Yt = X −

Z t

T

Zstr dBs



T

Z

Fbs (Zs )ds + KT − Kt ,

t ∈ [0, T ],

t

PH -q.s.,

(4.23)

bt ) for F given by (4.22). Using (4.19) and the representation formula in where Fbt (·) := Ft (·, a Proposition 4.9, we show the following

Theorem 4.20. 1. If X ∈ L2H and (Y, Z) ∈ D2H × H2H is a solution to the 2BSDE (4.23), then the good-deal bound is uniquely given by πtu (X) = Yt , t ∈ [0, T ], PH -q.s. and satisfies πtu (X) = Yt =

P

ess sup

P0

ess sup EtQ [X],

P 0 ∈PH (t+ ,P ) Q∈Qngd (P 0 )

t ∈ [0, T ], P -a.s. for all P ∈ PH .

2. For X ∈ L2H , there exists a unique solution (Y, Z) ∈ D2H × H2H to the 2BSDE (4.23). Proof. For z ∈ Rn , t ∈ [0, T ], the generator F (t, z, a) writes explicitly for a ∈ S>0 n ∩ [a, a] as  1/2



F (t, z, a) = −ht z tr a − aσttr (σt aσttr )−1 σt a) z

.

First we need to show that the function F (t, z, ·) : Rn×n → R is convex on its domain DFt = S>0 n ∩ [a, a], from which the Fenchel-Moreau theorem would imply that F (t, z, ·) is the convex conjugate of a nonlinear function H such that (4.3) holds. For this purpose, it suffices to show that the function Gt : a 7→ aσttr (σt aσttr )−1 σt a is S>0 n -convex. Let then µ ∈ [0, 1] >0 and a, a ˜ ∈ Sn . Using the Schur complement condition for positive semi-definiteness [HJ12, Theorem 7.7.7 or Theorem 7.7.16], convexity of Gt is equivalent to positive semi-definiteness of the matrix At ∈ R(n+d)×(n+d) given by 

At = 

tr 

µ aσttr (σt aσttr )−1 σt a + (1 − µ) a ˜σttr (σt a ˜σttr )−1 σt a ˜

σt (µa + (1 − µ)˜ a)



=µ

aσttr (σt aσttr )−1 σt a σt a

=: µA1t + (1 − µ)A2t .



σt (µa + (1 − µ)˜ a)σttr

σt (µa + (1 − µ)˜ a) tr 

σt a

σt aσttr



 + (1 − µ) 

a ˜σttr (σt a ˜σttr )−1 σt a ˜ ˜ σt a

tr 

σt a ˜

σt a ˜σttr



Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 162

Now since σt aσttr and σt a ˜σttr are positive definite and the set of positive semi-definite matrices is a convex cone, then the Schur complement condition applied to A1t and A2t implies that At is positive semi-definite. For existence and uniqueness of the solution to the 2BSDE (4.23) we aim to apply part 2 of Proposition 4.9. To this end, we show that F satisfies parts (i)-(v) of Assumption 4.6. Part (i) is clear by definition of F in (4.22) and the fact that DFt = S>0 n ∩ [a, a]. As for part (ii), it holds from the progressive measurability of the processes σ and h. To show that part (iii) about uniform continuity of F holds, recall that the point-wise product of two bounded uniformly continuous functions is uniformly continuous, and that the composition of two uniformly continuous functions is also uniformly continuous. With this it follows that F is uniformly continuous in ω with respect to k · k∞ for fixed (t, z, a) ∈ [0, T ] × Rn × Rn×n , since h and σ are uniformly continuous and bounded, σσ tr and σaσ tr are uniformly elliptic and bounded in the matrix norm, and the square root function is uniformly continuous. Since Fb 0 = 0, then part (v) obviously holds. It remains to show part (iv) about the Lipschitz continuity of Fb in z. By the Minkowski inequality one has PH -q.s. for all t ∈ [0, T ], that

1

1



1

1

b ⊥ 2 b ⊥ 2 0 0 2 b⊥ a b 2 (z − z 0 ) b t z − Πt a bt z ≤ ht Π |Fbt (z) − Fbt (z 0 )| = ht Π t a t bt (z − z ) ≤ khk∞ a t 











holds. Hence part (iv) follows and this concludes that F satisfies Assumption 4.6. 0

Part 1: Recall from the discussion preceding the statement of the theorem that π·u,P (X) for bt ), t ∈ P 0 ∈ PH (t+ , P ) and P ∈ PH solves the standard BSDE with generator Fbt (·) = Ft (·, a [0, T ], under P , for F given by (4.22). Part 1 is now a direct consequence of part 1 of Proposition 4.9, and the definition (4.19) of the good-deal bound π·u (X). Part 2: Direct application of part 2 of Proposition 4.9 gives the claim. Remark 4.21. 1. Were the dynamics (4.16) of the stock price processes rather given by the SDE dSt = diag(St )(bt dt + σt dBt ), with non-zero drift b, a candidate for the generator of the 2BSDE (4.23) would have been by Theorem 3.15 in Chapter 3 given for t ∈ [0, T ], z ∈ Rn , a ∈ Rn×n as F (t, z, a) =

( 2 1/2 a,⊥ 1  1  Πt a2 z ξta tr Πat a 2 z − h2t − ξta +∞

if a ∈ S>0 n ∩ [a, a] otherwise,

(4.24)

with ξta := (σt a1/2 )tr (σt aσttr )−1 bt ∈ Im (σt a1/2 )tr being the market price of risk in a model with volatility σa1/2 . Clearly this involves an additional dependence of F in a ∈ S>0 n ∩ [a, a], for which it becomes very difficult to see whether F is convex or not in a ∈ S>0 n ∩ [a, a]. Indeed a sufficient condition for the convexity of F (t, z, ·) given by (4.24) is that each summand is convex in a ∈ S>0 n ∩ [a, a]. However, the second summand of F is a product of two functions, and would be convex if the two components of the product are convex and either monotone increasing or monotone decreasing functions in

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 163

a ∈ S>0 n (cf. [BV04, Exercise 3.32]). But we have not been able to verify these properties and also the Schur complement condition is no longer enough to show convexity of the product. For these technical reasons we have modelled S directly as local martingale measures under P ∈ PH , i.e. with zero drift b = 0 (cf. (4.16) and Remark 4.18). 2. Theorem 4.20 shows in particular that the family of essential supremums in (4.19) indexed by the measures P ∈ PH effectively aggregates into a single process π·u (X). In fact using r.c.p.d. π·u (X) can be constructed without exception of a null-set, for X ∈ UCb (Ω) and then extended by density to X ∈ L2H (see [STZ13]). Moreover [STZ13, Proposition 4.11] implies that π·u (X) is actually F-progressively measurable, for X ∈ L2H . Hence by the Blumenthal Zero-One law (cf. Lemma 4.2) π0u (X) is constant and given by π0u (X) = supP ∈PH π0u,P (X). 3. By [STZ13, Proposition 4.7], the good-deal bound π·u (·) satisfies a dynamic programing principle (recursiveness): for all s ≤ t ≤ T , X ∈ L2H , holds P -a.s. for all P ∈ PH that πsu (X) =

P

ess sup

P 0 ∈PH (s+,P )

0

πsu,P (πtu (X)) =

P

ess sup

P0

ess sup EsQ [πtu (X)] = πsu (πtu (X)).

P 0 ∈PH (s+,P ) Q∈Qngd (P 0 )

This is equivalent to a time consistency property of the process π·u (X), for X ∈ L2H . 4. Using Proposition 4.12, it holds analogously to Lemma 3.1 in Chapter 3 (see also [KS07b, Theorem 2.7] or [Bec09, Proposition 2.6]) that the good-deal bound π·u (X) satisfies the properties of dynamic coherent risk measures (with generalized scenarios consisting of measures that can be associated to volatility uncertainty). In addition by part 2. it is time-consistent. These facts will be used to define good-deal hedging in terms of minimization of a risk measure of the type of π·u (·). We refer to [NS12] for a general study of dynamic risk measures under volatility uncertainty. Note that our subsequent results on hedging are, differently from [NS12], not on superhedging. Remark 4.22. We are not able to give more general examples of elements in L2H than those provided in part 3 of Remark 4.10. This is restrictive for financial applications where one would typically be interested in X being contingent claims that have some exponential dependence in + BT and hBiT , e.g. X = K − exp BT − hBiT /2 ∈ L2H in dimension n = 1 modeling a put option with strike K > 0 on a Black-Scholes risky asset with uncertain volatility. Clearly, this Markovian claim does not fit into the examples given in part 3 of Remark 4.10. Fortunately for some 2BSDE generators one can sometimes identify the solution to the 2BSDE via PDE arguments, even if X ∈ L2H does not belong to L2H ; cf. e.g. Section 4.3.3.

4.3.2

Robust good-deal hedging under volatility uncertainty

Our aim now is to define and characterize the good-deal hedging strategy using solutions to 2BSDEs. Here the objective of the investor is to find a PH -permitted trading strategy that

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 164

minimizes her residual risk (measured under some risk measure ρ) from any time onward when holding a liability X and trading dynamically in the market. Since the investor (say the seller) requires the premium π·u (X) for X, then she would like the good-deal valuation to be the minimal capital requirement to make her position acceptable. In this sense, the good-deal bound would be the market consistent risk measure associated to good-deal hedging via ρ; cf. [BE09]. The risk to be minimized is measured in terms of a dynamic risk measure compatible with the no-good-deal constraint in the market and the uncertainty-aversion of the investor. The second objective of the investor should be towards robustness (of hedges and valuations) with respect to volatility uncertainty. As in Proposition 3.25 of Chapter 3 we show robustness of the good-deal hedging strategy as a supermartingale property of its tracking (hedging) error with respect to a class of a-priori valuation measures P ngd ⊇ ∪P ∈PH Qngd (P ), i.e. uniformly over all reference models P ∈ PH . Recalling the definition of π·u (X) (for X ∈ L2H ) in (4.19) and previous results on good-deal valuation and hedging in the absence of model uncertainty (cf. e.g. [Bec09, Theorem 5.4] or Theorem 3.17 in Chapter 3), one has for all P ∈ PH , and P 0 ∈ PH (t+ , P ) P

0

0



πtu,P (X) = ess inf0 ρPt X −

T

Z

φ∈Φ(P )

t



P -a.s., t ∈ [0, T ],

φtr s dBs ,

(4.25)

P

where for P ∈ PH we define ρPt (X) := ess sup EtQ [X], P -a.s., t ∈ [0, T ] with Q∈P ngd (P )

n

o

P ngd (P ) := Q ∼ P | dQ/dP = (P )E(λ · W P ), λ progressively measurable, |λ| ≤ h . Here P ngd (P ) is the set of a-priori valuation measures equivalent to P which satisfy the no-good-deal restriction under P , but might fail to be local martingale measures for the stock price process S (yet they are with respect to the trivial market with only the riskless asset S 0 ≡ 1). In particular for each P ∈ PH , the set P ngd (P ) is also m-stable and convex. This implies that the dynamic coherent risk measure ρP : L2 (P ) → L2 (P, Ft ) is time-consistent (see e.g. Lemma 3.1 in Chapter 3) satisfying ρP· (X) ≥ π·u,P (X) since P ngd (P ) ⊇ Qngd (P ). Furthermore from (4.21) and (4.25), we have for all t ∈ [0, T ] and P ∈ PH that πtu (X) =

P

P

ess sup

0



ess inf0 ρPt X −

P 0 ∈PH (t+ ,P ) φ∈Φ(P )

Z t

T



φtr s dBs , P -a.s..

(4.26)

In addition for X ∈ L2H it can be inferred n from [Bec09, Theorem o 5.4] (see also Theorem 3.17 in P ¯ Chapter 3) that there exists a family φ ∈ Φ(P ), P ∈ PH of trading strategies satisfying πtu (X)

=

P

ess sup

P 0 ∈PH (t+ ,P )

0 ρPt



X−

Z t

T

0



(φ¯Ps )tr dBs , P -a.s., for all t ∈ [0, T ], P ∈ PH . (4.27)

Moreover φ¯P is given for P ∈ PH by  1/2 ¯P b b1/2 Z P,X , t ∈ [0, T ], P -a.s., bt φ a t = Πt a t t

(4.28)

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 165

where (Y P,X , Z P,X ), P ∈ PH , is the solution to the standard BSDE under P with terminal bt ), t ∈ [0, T ], for F defined in (4.22), satisfying condition X and generator −Fbt (·) = −Ft (·, a u,P P,X Y = π· (X). If PH were a singleton PH = {P }, then for X ∈ L2H = L2 (P ) the strategy P φ¯ would be PH -permitted and hence already the solution to the good-deal hedging problem with the valuation π·u (X) = π·u,P (X) associated to the risk measure ρP . In the present non-dominated framework however, the situation is more subtle because the strategies φ¯P and risk measures ρP may be defined only up to a null-set of the associated probability measure P ∈ PH . Since we are looking for a PH -permitted hedging strategy, one way is to investigate appropriate conditions under which the family {φ¯P , P ∈ PH } can be aggregated into a single strategy φ¯ ∈ Φ, i.e. φ¯ = φ¯P P ⊗ dt-a.s., for any P ∈ PH . If this were possible, then (4.27) would write πtu (X) =

P

0

ess sup

P 0 ∈PH (t+ ,P )



= ρt X −

Z t

T



ρPt X −

Z t

T



φ¯tr s dBs , t ∈ [0, T ], P -a.s. for all P ∈ PH



φ¯tr s dBs , t ∈ [0, T ], P -a.s. for all P ∈ PH ,

where ρ· (X) ∈ D2H is defined for X ∈ L2H as the unique process (if it exists) that satisfies ρt (X) =

P

ess sup

P 0 ∈PH (t+ ,P )

0

ρPt (X), t ∈ [0, T ], P -a.s. for all P ∈ PH .

(4.29)

As general conditions for aggregation (see e.g. [STZ11]) can be somewhat restrictive and technical, we will express the hedging strategy in terms of the control component Z of the unique solution (Y, Z) to the 2BSDE (4.23). Note that even in case there exists a worst-case ¯ measure P¯ ∈ PH such that ρ = ρP , it is not not clear at all whether a hedging strategy in the model P¯ is robust with respect to all measures in P ngd (P ) for any P ∈ PH , in the sense that the supermartingale property of tracking errors holds uniformly under any Q ∈ ∪P ∈PH P ngd (P ). An analogous issue was already noticed in Subsection 3.3.4 of Chapter 3 under drift uncertainty. The issue was addressed there by first considering a larger valuation bound for which a robust hedging strategy uniformly with respect to all priors exists, i.e. a strategy that satisfies a supermartingale property of tracking error under all measures a-priori valuation measure uniformly over all priors. A subsequent step was then to identify this larger bound with the standard good-deal valuation bound. Here relying on the intuition from Theorem 3.28 and Theorem 3.30 in Chapter 3, we can write down what a candidate hedging strategy (cf. (4.34)) in our setup in terms of the solution to the 2BSDE (4.23). From this we can then proceed in a more straightforward manner to show directly that this candidate strategy is indeed a good-deal hedging strategy and that it satisfies the required robustness property with respect to uncertainty. Clearly ρ is a dynamic coherent risk measure analogous to π·u (X). The good-deal hedging problem under volatility uncertainty consists in minimizing over PH -permitted trading strategies

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 166

the dynamic residual risk measured under ρ. This is done in such a way that at every time the minimal capital required for acceptability coincides with the good-deal valuation bound. More precisely for a contingent claim X ∈ L2H , we aim to find φ¯ ∈ Φ such that for all t ∈ [0, T ] and P ∈ PH holds πtu (X)

P



= ess inf ρt X −

Z

φ∈Φ

t

T

φtr s dBs





= ρt X −

Z

T

t



φ¯tr s dBs , P -a.s..

(4.30)

To introduce the notion of robustness with respect to volatility uncertainty, recall the definition of the tracking error Rφ (X) of a permitted strategy φ ∈ Φ for a claim X ∈ L2H : Rtφ = πtu (X) − π0u (X) −

t

Z 0

φtr s dBs ,

t ∈ [0, T ], P -a.s. for all P ∈ PH .

(4.31)

In other words, the tracking error is the difference between the dynamic variations in the capital requirement and the profit or loss from trading. As in Subsection 3.3.3 of Chapter 3, ¯ we will say that a good-deal hedging strategy φ(X) for a claim X is robust with respect to ¯ φ uncertainty if R (X) is a supermartingale under every measure Q ∈ P ngd (P ) uniformly for all P ∈ PH . Again as in Chapter 3, this means that a robust hedging strategy φ¯ is at least mean-self-financing uniformly over all Q ∈ ∪P ∈PH P ngd (P ). Let us make a short transit and provide a 2BSDE description of the risk measure ρ· (X), for X ∈ L2H . As in part 4. of Remark 4.21, this yields in particular time-consistency of the dynamic risk measure ρ over contingent claims X in L2H . For this purpose, define the function F 0 : Ω × [0, T ] × Rn × Rn×n → R by  −h |a1/2 z|, t F 0 (t, z, a) = +∞

if a ∈ S>0 n ∩ [a, a], otherwise.

(4.32)

Consider the 2BSDE Yt0 = X −

Z t

T

tr

Z 0 s dBs −

Z

T

t

Fbs0 (Zs0 )ds + KT0 − Kt0 ,

t ∈ [0, T ], PH -q.s.,

(4.33)

with generator F 0 defined in (4.32). Proposition 4.23. 1. If X ∈ L2H and (Y 0 , Z 0 ) ∈ D2H × H2H is a solution to the 2BSDE (4.33), then ρ· (X) is uniquely given by ρt (X) = Yt0 , t ∈ [0, T ], PH -q.s. and satisfies ρt (X) = Yt0 =

P

ess sup

P0

ess sup EtQ [X],

P 0 ∈PH (t+ ,P ) Q∈P ngd (P 0 )

t ∈ [0, T ], P -a.s. for all P ∈ PH .

2. For X ∈ L2H , there exists a unique solution (Y 0 , Z 0 ) ∈ D2H × H2H to the 2BSDE (4.33).

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

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0 tr 1/2 and using again the Proof. Rewriting F 0 on a ∈ S>0 n ∩ [a, a] as F (t, z, a) = −ht (z az) Schur complement condition (see proof of Theorem 4.20), one also proves that that F 0 is a convex function of a for fixed (t, z) ∈ [0, T ] × Rn . This by Fenchel-Moreau theorem implies that F 0 is the convex conjugate of a nonlinear function H 0 such that an analog of (4.3) holds. In addition, it is easy to verify as in the proof of Theorem 4.20 that F 0 satisfies Assumption 4.6.

Part 1: By [Bec09], it is known that Y˜tP,X = ρPt (X), P -a.s., t ∈ [0, T ], where (Y˜ P,X , Z˜P,X ) denotes the unique solution to the standard Lipschitz BSDE under P with generator −Fb 0 and terminal condition X, for P ∈ PH . Hence part 1 is also a direct consequence of part 1 of Proposition 4.9 and the definition of ρ in (4.29). Part 2: As a consequence of part 2 of Proposition 4.9, for X ∈ L2H the 2BSDE (4.33) admits a unique solution (Y 0 , Z 0 ) ∈ D2H × H2H . We characterize φ¯ in terms of the unique solution (Y, Z) of the 2BSDE (4.23) and show that it is robust with respect to volatility uncertainty. Using the intuition from robust hedging in the presence of drift uncertainty (see Theorem 3.28 and Theorem 3.30 in Chapter 3), a candidate ¯ good-deal hedging strategy for X ∈ L2H is φ¯ := φ(X) defined by 1/2 ¯ b b1/2 Zt ), bt φ a t := Πt (a t

t ∈ [0, T ], PH -q.s.,

(4.34)

where (Y, Z) is a solution to the 2BSDE (4.23). Since Z is already defined PH -quasi-surely ¯ in (4.34) is also defined PH -quasi-surely and it b is defined pathwise, then the strategy φ and a can be shown it is indeed a robust good-deal hedging strategy if the “gain/loss” family of n that o R · tr (P ) processes 0 Zt dBt , P ∈ PH aggregates. The precise result is the following 2 is a solution to the 2BSDE (4.23) Theorem 4.24. Assume X ∈ L2H and that (Y, Z) ∈ D2H ×H nH o

with generator F given by (4.22) such that the integrals into a single process



tr 0 Zt dBt

R (P ) · Z tr dB , t 0 t

P ∈ PH , aggregate

(equivalently 2BSDE (4.23) admits a solution (Y, Z, K)). Then:

¯ 1. The strategy φ¯ = φ(X) given by (4.34) is in Φ and solves the good-deal hedging problem under uncertainty (4.30). ¯

¯ 2. The tracking error process Rφ (X) of the hedging strategy φ¯ = φ(X) is a supermartingale ngd under any Q in ∪P ∈PH P (P ). Proof. We first prove part 2., since the proof of part 1. will use it. By Theorem 4.20, we know that π·u (X) = Y for (Y, Z) solution to the 2BSDE (4.23). Let P ∈ PH and Q ∈ P ngd (P ). Then ¯ ¯ Q is equivalent to P and dQ = (P )E(λ · W P )dP for |λ| ≤ h. The dynamics of Rφ := Rφ (X)

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 168

is then given under P by ¯ P −dRtφ = −Fbt (Zt )dt − Zttr dBt + φ¯tr t dBt + dKt ,

P -a.s.,

1 2

bt dWtP + dKtP , = −Fbt (Zt )dt − (Zt − φ¯t )tr a

P -a.s.,

for all t ∈ [0, T ], with {K P , P ∈ PH } the non-decreasing adapted processes defined as in R (4.7). Changing measures to Q for the Q-Brownian motion W Q = W P − 0· λt dt gives for t ∈ [0, T ] that ¯

1

1

¯t ))dt − (Zt − φ¯t )tr a bt2 dWtQ + dKtP , bt2 (Zt − φ −dRtφ = (−Fbt (Zt ) − λtr t a

P -a.s.

holds. Now with (4.22) and the expression of φ¯ in (4.34) one rewrites P -a.s. for t ∈ [0, T ] ¯ −dRtφ

  1 1 ⊥ 1 tr b ⊥ 2 2 b = ht Πt (b at Zt ) − λt Πt (b at Zt ) dt − (Zt − φ¯t )tr b at2 dWtQ + dKtP . 1

1

2 b ⊥ b 2 Zt ) = ht Π b ⊥ (a Since K P is non-decreasing and max|λt |≤ht λtr t Πt (a t bt Zt ) , then the finite t ¯ ¯ variation part of the Q-semimartingale Rφ is non-increasing. Note that Rφ ∈ S 2 (P ) since p π·u (X) ∈ D2H ⊂ S 2 (P ) and φ¯ ∈ Φ(P ). Finally, since λ is bounded, dQ dP is in L (P ) for ¯ any p < ∞ and by H¨older’s inequality it follows that Rφ ∈ S 2−ε (Q) (ε > 0) holds. As a ¯ consequence Rφ is clearly a Q-supermartingale.





Now to prove part 1. note first that by the condition on the integral Z, the strategy φ¯ given by (4.34) belongs to Φ. Now to show that φ¯ solves the hedging problem (4.30), let P ∈ PH , and P 0 ∈ PH (t+ , P ). Then for any φ ∈ Φ it holds φ · B is a Q-martingale in S 1 (Q) for any Q ∈ Qngd (P 0 ) since the Girsanov kernels of measures Q with respect to P 0 are all uniformly R 0 0 bounded. Because Qngd (P 0 ) ⊆ P ngd (P 0 ), this implies that πtu,P (X) = πtu,P (X− tT φtr s dBs ) ≤ R T tr 0 P 0 0 + ρt (X − t φs dBs ), P -a.s.. Taking the essential supremum over P ∈ PH (t , P ) first and R then the essential infimum over φ ∈ Φ yields πtu (X) ≤ ess infPφ∈Φ ρt (X − tT φtr s dBs ), P -a.s.. ¯ Hence to show that φ is a good-deal hedging strategy satisfying (4.30), it suffices to show   R ngd (P 0 ) and P 0 ∈ P (t+ , P ). that πtu (X) ≥ EtQ X − tT φ¯tr H s dBs , P -a.s. holds for all Q ∈ P 0 + ngd 0 To this end, let P ∈ PH (t , P ) and Q ∈ P (P ). From part 1. of the theorem, the ¯ ¯ supermartingale property of the tracking error Rφ := R·φ (X) of φ¯ under Q implies that   R R Q T ¯tr u πtu (X) − π0u (X) − 0t φ¯tr s dBs ≥ Et X − π0 (X) − 0 φs dBs . Reorganizing the last inequality yields the claim. In general, the Itˆo’s stochastic integrals of the form 0· Zttr dBt are only defined P -almost surely under each P ∈ PH . As already mentioned before, sufficient conditions for aggregation of R processes can be quite restrictive. By a result of [Kar95] it is possible to define 0· Zttr dBt pathwise, and in particular such that it satisfies the hypothesis of Theorem 4.24, if the process Z is c`adl`ag. Note that in our setup the Z-component of a 2BSDE solution (Y, Z) does not have R

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 169

to be a c`adl`ag process in general. We emphasize however that this does not make Theorem 4.24 totally inapplicable. Indeed under some Markovian assumptions one may sometimes be able to use PDE arguments to show that the Z-component is c`adl`ag. An example of such a situation is provided in Section 4.3.3 below, where we obtain explicit solutions to the 2BSDE (4.23), for some contingent claim satisfying X ∈ L2H and probably not X ∈ L2H . In a general context, a R result of [Nut12b] shows that the stochastic integral 0· Zttr dBt can be defined pathwise for any predictable process Z if one complements the Zermelo-Fraenkel axioms of set theory (which are by now well-accepted) with the combination “continuum hypothesis plus the axiom of choice” or the softer one “negation of the continuum hypothesis plus the so-called Martin’s axiom” (see [DM78, Chapter II, Sections 27-29]). Note under the conditions of [Nut12b] that for a solution (Y, Z) of a 2BSDE for which Y is PH -quasi-surely defined, the family {K P , P ∈ PH } will automatically aggregate into a single process K such that (Y, Z, K) becomes a solution to the 2BSDE. As for the 2BSDE (4.23) of interest in Theorem 4.24, we already know by Theorem 4.20 that this would be the case if X ∈ L2H . As a further remark, note that Part 2. of Theorem 4.24 can be interpreted as a robustness property of the good-deal hedging strategy φ¯ with respect to volatility uncertainty. Finally, a direct consequence of Theorem 4.24 (when its conditions are satisfied) is the following minmax identity: for all t ∈ [0, T ] , P ∈ PH one has by (4.26) and (4.30) that π ¯tu (X)

P

:= ess inf φ∈Φ

=

4.3.3

P

ess sup

P 0 ∈PH (t+ ,P ) P

ess sup

P

0 ρPt



 0

X−

ess inf0 ρPt X −

P 0 ∈PH (t+ ,P ) φ∈Φ(P )

T

Z t

Z t

T

φtr s dBs





u φtr s dBs = πt (X), P -a.s.,

Example for options on non-traded assets

We provide an example for robust good-deal valuation and hedging of European put options on a non-traded asset under volatility uncertainty. The financial market consists of a traded stock of Black-Scholes’ type with (discounted) price process S and a non-traded asset with value process L. Hence d = 1 and n = 2 for the framework of Section 4.2. For the canonical process B = (B 1 , B 2 ), the set PH of local martingale measures is defined as in (4.1) via constant diagonal matrices a, a ∈ S>0 2 given by a = diag(a1 , a2 ) and a = diag(a1 , a2 ), such b ≤ a, PH ⊗ dt-q.s.. We model (S, L) as that a ≤ a dSt = St σ S dBt1

and dLt = Lt γdt + β(ρdBt1 +

q

1 − ρ2 dBt2 ) , 

PH -q.s.,

with S0 , L0 > 0, a volatility matrix σ := (σ S , 0) ∈ R1×2 of maximal rank 1 = d < n = 2, σ S , β ∈ (0, ∞), γ ∈ R, and P 0 -correlation coefficient ρ ∈ [−1, 1] for returns of S and L. For a constant bound h ∈ [0, ∞) on the instantaneous Sharpe ratios, we derive explicit formulas

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 170

for the worst-case good-deal valuation bound and robust hedging strategy of European put options X = (K − LT )+ , with strike K ∈ R+ and maturity T . We denote !

b11 a b12 a b12 a b22 a

b= a

1 2

b = and a

b b11

b b12

b b12

b b22

!

,

b being the S>0 bt dt pathwise and a ≤ a b ≤ a, PH ⊗dt-q.s. for a 2 -valued process satisfying dhBit = a as in (4.1). One has b11 = (b a b11 )2 + (bb12 )2 ,

b12 = b a b12 (bb11 + bb22 ),

b22 = (b a b22 )2 + (bb12 )2 ,

(4.35)

2 b11 a b22 − (a b12 )2 = b and a b11bb22 − (bb12 )2 . 1

b 2 = σ S (b Since σ a b11 , bb12 ), then from their respective definitions hold  1 b 2 )tr = z ∈ R2 : b Im (σ a b12 z1 − bb11 z2 = 0

 1 b 2 ) = z ∈ R2 : b and Ker (σ a b11 z1 + bb12 z2 = 0 ,

which imply for z ∈ R2 that !

(bb11 )2 z1 + bb11bb12 z2 b b11bb12 z1 + (bb12 )2 z2

1 b Π(z) = 11 b a

(bb12 )2 z1 − bb11bb12 z2 (bb11 )2 z2 − bb11bb12 z1

b ⊥ (z) = 1 and Π b11 a

!

.

(4.36)

˜ there satisfies Clearly LT ∈ L2H follows from the estimate (4.44) below, since the process L P ˜ Et [L] ≤ 1 for any t ∈ [0, T ], P ∈ PH . In addition since the put option payoff function x 7→ (K − x)+ is bounded and Lipschitz continuous, it follows that X = (K − LT )+ ∈ L2H . Recall from (4.21) that the worst-case good-deal bound πtu (X) (if it exists) for X for t ∈ [0, T ] satisfies for any P ∈ PH πtu (X) =

P

ess sup

P 0 ∈PH (t+ ,P )

0

πtu,P (X) =

P

ess sup

P

ess sup EtQ [X], P -a.s..

P 0 ∈PH (t+ ,P ) Q∈Qngd (P 0 )

b⊥ a b11 a b22 − (a b1/2 z = a b12 )2 From (4.36) and using (4.35), follows Π solution to the 2BSDE (4.23) which rewrites here as

Yt = X −

Z

T

Fb (s, Zs )ds −

t

Z t

T



1/2

b11 a

−1/2 z2 . is a

Zstr dBs + KT − Kt , t ∈ [0, T ], PH -q.s.,

(4.37)

1/2

−1/2

z2 , with generator given by F (t, z, a) = −h Πa,⊥ a1/2 z = −h a11 a22 − (a12 )2 a11 >0 tr 2 for a ∈ S2 ∩ [a, a] and F (t, z, a) = +∞ otherwise, for z = (z1 , z2 ) ∈ R . We show in Lemma 4.25 below that the solution to the BSDE (4.37) is given by

Yt = v(t, Lt )

and Zt = βLt



q tr ∂v (t, Lt ) ρ, 1 − ρ2 , PH -q.s. ∂x

hold for every t ∈ [0, T ], where v ∈ C 1,2 [0, T ) × (0, ∞), R is the classical solution to the Black-Scholes’ type PDE ( p √  ∂v 1 2 2 ∂v ∂2v + γ − hβ 1 − ρ2 a2 x ∂x + 2 β ρ a1 + (1 − ρ2 )a2 ) x2 ∂x 2 = 0 ∂t (4.38) v(T, LT ) = (K − LT )+ . 

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 171

We need some preparations towards this result. Let P a = P 0 ◦ (a1/2 B)−1 ∈ PH be the local martingale measure satisfying hBit = at, P a -a.s., for all t ∈ [0, T ]. The dynamics of the process L under P a is the geometric Brownian motion q   √ √ a a dLt = Lt γdt + β ρ a1 dWt1,P + 1 − ρ2 a2 dWt2,P , t ∈ [0, T ], a which can be rewritten as dLt = Lt γdt + β¯ ρ¯dWt1,P +





β¯ := β ρ2 a1 + (1 − ρ2 )a2

1/2

>0

1 − ρ¯2 dWt2,P

p

a



, t ∈ [0, T ], for

−1/2 √  and ρ¯ := ρ a1 ρ2 a1 + (1 − ρ2 )a2 ∈ [−1, 1],

where W P = (W 1,P , W 2,P ) = (a) −1/2 B is a P a -Brownian motion. The Black-Scholes formula applied for the dynamics of L under P a provides a closed-form expression for v(t, Lt ), for v solution to the PDE (4.38). Using arguments analogous to the ones in the derivations of (3.31) in Section 3.2.2 of Chapter 3 it can be shown that v(t, Lt ) coincides with the good-deal a

a

a

a

valuation bound πtu,P (K − LT )+ in the model under P a . Furthermore, an explicit formula for both is given by K 

a

v(t, Lt ) = πtu,P (X) (4.39)

= KN (−d− ) − Lt em(T −t) N (−d+ ) =em(T −t) ∗ B/S-put-price time: t, spot price: Lt , strike: Ke−m(T −t) , vol: β¯ , 

with “ B/S-put-price” being the standard Black-Scholes formula for interest rate being zero, with “vol” being the argument for volatility in the Black-Scholes model, where q q √ 2 ¯ m := γ − hβ 1 − ρ¯ = γ − hβ 1 − ρ2 a2 ,   −1  d± := ln Lt /K + m ± 12 β¯2 (T − t) β¯ (T − t) and N is the cumulative distribution function of the standard normal law. The following lemma identifies the solution to the 2BSDE (4.37) via the solution v of the PDE (4.38). p

Lemma 4.25. The solution (Y, Z, K) of the 2BSDE (4.37) is given by Yt = v(t, Lt ), Zt = p tr ∂v βLt ∂x (t, Lt ) ρ, 1 − ρ2 and K given by (4.41), for t ∈ [0, T ], with (Y, Z) ∈ D2H × H2H R such that the stochastic integral 0· Zttr dBt is pathwise defined. Proof. For any P ∈ PH , applying Itˆo’s formula and using (4.38) yields for t ∈ [0, T ] v(t, Lt ) = X −

Z t

T

Zstr dBs

Z

+h t

T

2 b11 b22 b12 a s a s − (a s )

1/2

b11 a s

−1/2 2 Z ds + KT − Kt , P -a.s., s

with Z = (Z 1 , Z 2 )tr given from (4.39) by q q tr tr ∂v m(T −t) 2 Zt = βLt (t, Lt ) ρ, 1 − ρ = −βe N (−d+ )Lt ρ, 1 − ρ2 ∂x

(4.40)

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 172

and Z t

Kt :=

 √  ∂v 22 12 2 1/2 11 −1/2 b11 b b b (4.41) a2 (s, Ls ) a a − ( a ) a − s s s s ∂x   p 1 2 2 ∂2v 2 11 2 22 12 bs ) + (1 − ρ )(a2 − a bs ) − 2ρ 1 − ρa bs + β Ls 2 (s, Ls ) ρ (a1 − a ds. 2 ∂x q

hβ 1 − ρ2 Ls

0

b ≤ a P ⊗ dt-a.s. yields a b1/2 ≤ To show that K is a non-decreasing process, note that a a1/2 P ⊗ dt-a.s. and both inequalities imply that b11 ) + (1 − ρ2 )(a2 − a b22 ) − 2ρ 1 − ρ a b12 ρ2 (a1 − a p

= ρ,

q

1 − ρ2

tr

a ρ,

q



1 − ρ2 − ρ,

and b11 a b22 − (a b12 )2 a

1/2

b11 a

q

−1/2

1 − ρ2

tr

b22 ≤ a

b ρ, a

1/2



q



1 − ρ2 ≥ 0

√ a2

(4.42)

(4.43)

hold P ⊗ dt-almost surely. Thus the process K is P -a.s. non-decreasing, because the delta of the put option in the Black-Scholes model is non-positive and the gamma is non-negative, ∂v ∂2v i.e. ∂x (t, Lt ) ≤ 0 and ∂x 2 (t, Lt ) ≥ 0 for all t ∈ [0, T ] using (4.39). Moreover the process K satisfies the minimum condition (4.8). This can be shown following arguments analogous to those in the proof of [STZ12, Theorem 5.3]; we reproduce the arguments for the convenience of the reader. Indeed, let us define l : (0, ∞) × R2 → R basically as the generator function (minus the γ-term) of the PDE (4.38) defined for (x, p, q) ∈ R+ × R2 by q √ 1 l(x, p, q) := −hβ 1 − ρ2 a2 xp + β 2 ρ2 a1 + (1 − ρ2 )a2 ) x2 q, 2

so that K =



kt = l Lt ,

0 ks ds

holds with

  p  1 ∂v 2 22 12 b b b11 b + (1 − ρ ) a + 2ρ (t, Lt ), Γt − β 2 L2t ρ2 a 1 − ρ a t t t Γt + F (t, Zt ), ∂x 2

∂v ∂2v (t, Lt ) and Zt given by (4.40). Since l Lt , ∂x (t, Lt ), Γt is by (4.42) and (4.43) the ∂x2   √ 1 2 2 2 11 2 22 12 supremum of 2 β Lt ρ at + (1 − ρ )at + 2ρ 1 − ρat Γt − Fb (t, Zt ) over a ∈ DF := [a, a], then by measurable selection arguments there exists for every  > 0 a predictable process a

for Γt =



valued in DF such that l Lt ,

  p  1 ∂v + (1 − ρ2 )a,22 + 2ρ 1 − ρa,12 Γt (t, Lt ), Γt ≤ β 2 L2t ρ2 a,11 t t t ∂x 2 − F (t, Zt , at ) + .

Now let P α ∈ PH and t0 ∈ [0, T ] be fixed, and define recursively the sequence (τn )n of random times τ0 := inf{t ≥ t0 : Kt ≥ Kt0 + } ∧ T , and n

 τn+1 := inf t ≥ τn :

l Lt ,

 ∂v (t, Lt ), Γt + F (t, Zt , aτn ) ≥ ∂x

 o p 1 2 2  2 ,11 ,12 β Lt ρ aτn + (1 − ρ2 )a,22 + 2ρ 1 − ρa Γ + 2 ∧ T.   t τn τn 2

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 173

Since K, L, Z, Γ are continuous, then τn is a F-stopping time for any n, and τ0 > t0 . Furthermore since l is continuous and F (·, a) also is for fixed a in DF , then for PH -quasi all ω the function l Lt ,

  p  1 ∂v ,12 2 ,22 + 2ρ (t, Lt ), Γt − β 2 L2t ρ2 a,11 + (1 − ρ )a 1 − ρa Γt + F (t, Zt , aτn )    τn τn τn ∂x 2

is uniformly continuous in t on the compact interval [τn , T ]. Hence uniformly over n holds  (ω) − τ  (ω) ≥ δ(, ω) > 0 whenever {τ  (ω) < T }, which implies τ  (ω) = T for large τn+1 n n n enough n. Now from the arguments in [STZ11, Example 4.5] applied to the interval [τ0 , T ], there exists a F-progressively measurable process α valued in DF such that α = α on [0, τ0 ]

b= and a

∞ X



α  aτn 1[τn ,τn+1 ⊗ dt-a.s. on Ω × [τ0 , T ]. ), P

n=0 

It follows that k ≤ 2, P α ⊗ dt-a.s. on Ω × [τ0 , T ], which implies for P := P α ∈ PH that 0≤

P

α

0

ess inf

P 0 ∈PH (t0 +,P )

EtP0 [KT − Kt0 ] ≤  + EtP0 [KT − Kτ0 ] ≤  + 2(T − t0 ), P -a.s.,



since P α ∈ P(t0 +, P ) because τ0 > t0 . Taking the limit as  tends to zero yields that K satisfies the minimum condition (4.8). It remains to show that v(·, L· ) ∈ D2H and Z ∈ H2H . This will conclude by uniqueness of the solution to the 2BSDE (4.23) (see Theorem 4.20) that (v(·, L· ), Z) for Z given in (4.40) is the unique solution to the 2BSDE (4.37). Since v is of class C 1,2 and L is PH -q.s. continuous, then v(·, L· ) and Z are F+ -progressively measurable. That v(·, L· ) is in D2H now follows from (4.39) which indeed implies that 0 ≤ v(t, Lt ) ≤ K holds pathwise. From (4.40) and since 1/2 2 b ≤ a holds P -a.s. for any P ∈ PH , one has a bt Zt ≤ max(a1 , a2 )β 2 e2|m|T L2t for a ≤ a all t ∈ [0, T ], P -a.s. for any P ∈ PH . Hence to show Z ∈ H2H it suffices to show that R  supP ∈PH E P 0T L2t dt < ∞. For this purpose, note that for any P ∈ PH it holds that T

Z 0

L2t dt

≤β

−2

−1

(min(a1 , a2 ))

hLiT

and

L2T





2 ˜T L20 e 2|γ|+β max(a1 ,a2 ) T L

(4.44)

˜ satisfying L ˜ = 1 + · 2L ˜ s β ρBs1 + 1 − ρ2 Bs2 , PH -q.s.. Clearly P -almost surely, for L 0 ˜ T ] ≤ 1 holds for every P ∈ PH , and thus taking expectations in (4.44) gives E P [L R

E

P

Z 0

T

 L2t dt

≤β

−2

p





2 (min(a1 , a2 ))−1 L20 e 2|γ|+β max(a1 ,a2 ) T ,

for all P ∈ PH .

Now taking the supremum over P ∈ PH implies the result. So (v(·, L· ), Z) is the unique R solution to the 2BSDE (4.37) in D2H × H2H . Finally that 0· Zttr dBt is pathwise defined follows from [Kar95] since Z is continuous and F+ -adapted.

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 174

Lemma 4.25 implies that πtu (K − LT )+ = v(t, Lt ) 

and Zt = Zt1 , Zt2

tr

= βLt

tr  q ∂v (t, Lt ) ρ, 1 − ρ2 , t ∈ [0, T ]. ∂x

Hence the robust good-deal bound πtu (K − LT )+ is attained for the largest “volatility matrix” a, and can be computed as in the absence of uncertainty, but under a worst-case   a measure P a ∈ PH for which hBit = at P a -a.s., yielding πtu (K − LT )+ = πtu,P (K − LT )+  for t ∈ [0, T ]. In addition, πtu (K − LT )+ is given explicitly by the Black-Scholes type formula (4.39), for modified strike price and volatility corresponding to K exp(−m(T − t)) and 1/2 respectively. Similarly, one can show that the lower good-deal β¯ = β ρ2 a1 + (1 − ρ2 )a2  l + bound πt (K − LT ) can be computed as in the absence of uncertainty, but under the a worst-case measure P ∈ PH corresponding to the lowest “volatility matrix” a. Furthermore, ¯ the robust good-deal hedging strategy φ¯ := φ(X) for the put option X = (K − LT )+ is given  1/2 ¯ b b1/2 Zt ), for Z = Z 1 , Z 2 tr given by (4.40), i.e. bt φ by a t = Πt (a t 

−1/2 b 1/2 bt bt ρ, φ¯t = −βem(T −t) N (−d+ )Lt a Πt a



q

1 − ρ2

tr 

,

for all t ∈ [0, T ], PH -q.s..

Now for a vector z = (z1 , z2 )tr ∈ R2 , straightforward calculations using (4.36) and (4.35) imply b a

−1/2 b

1/2

b Π a

1

b b22

b b11bb22 − (bb12 )2

−bb12



z =

· = =

 ! b b12 (bb11 )2 + bb11bb12bb22 z2  b b11 (bb12 )2 + bb22 (bb12 )2 z2  11  ! b b z1 + a b12 z2 b11bb22 − (bb12 )2 a 

(bb11 )2 + (bb12 )2 1 b b11bb22 − (bb12 )2 ! b a12 z1 + b z 2 11 a .

!

(bb11 )3 + bb11 (bb12 )2 z1 +  b b12 (bb11 )2 + (bb12 )3 z1 +

1

b11 a

−bb12 b b11



0

0

Hence an explicit formula for φ¯t is q  tr b12 a t m(T −t) ¯ φt = −βe N (−d+ )Lt ρ + 11 1 − ρ2 , 0 , for all t ∈ [0, T ], PH -q.s.. bt a

(4.45)

As the optimal growth rate bound h tends to infinity, the good-deal bound π·u (X) increases towards the robust upper no-arbitrage bound under volatility as studied in [ALP95, Lyo95, DM06, NS12, Vor14]. The put option X = (K − LT )+ being a claim with convex payoff function, our result agrees with those of [ALP95, Lyo95, EJPS98, Vor14] according to which in the presence of volatility uncertainty, no-arbitrage valuation of put options under maximal (resp. minimal) volatility corresponds to the worst-case for the seller (resp. buyer). The latter works focus on the robust super-replication problem under volatility uncertainty for valuation

Section 4.3. Good-deal bounds and hedging under volatility uncertainty

Page 175

with respect to the worst-case no-arbitrage bound. Here we instead study the robust gooddeal hedging problem under volatility uncertainty for valuation with respect to the worst-case good-deal bound. Let us also mention that [ALP95, EJPS98, Vor14] work in a one-dimensional model with a single risky asset and obtain as super-replicating strategy the delta of the option under the worst-case measure. This is included in our case study as a special case for |ρ| = 1. In a generalization towards a two-dimensional model, we consider possibly non-perfectly correlated (traded and non-traded) risky assets and derive a robust good-deal hedging strategy for the worst-case good-deal valuation in a market that is possibly incomplete under each fixed prior P ∈ PH . Furthermore the robust good-deal hedging strategy φ¯ here is not (the risky asset component of) the super-replicating strategy, in particular, when 0 < |ρ| < 1. Indeed since 0 ≤ X ≤ K holds pathwise, the no-arbitrage bound process Vb (X) under P a defined by Pa

Vbt (X) := ess sup EtQ [X], t ∈ [0, T ], P a -a.s. Q∈Me (P a )

a

a

a

satisfies πtu,P (X; h) ≤ Vbt (X) ≤ K, P a -a.s. for πtu,P (X; h) given by πtu,P (X) in (4.39). In a

addition if |ρ| < 1 then πtu,P (X; h) % K as h % +∞ (since then m → −∞, d± → −∞). These imply that if |ρ| < 1 then Vbt (X) = K1{t